The most outrageous (or ridiculous) conjectures in mathematics The purpose of this question is to collect the most outrageous (or ridiculous) conjectures in mathematics.
An outrageous conjecture is qualified ONLY if:
1) It is most likely false
(Being hopeless is NOT enough.)
2) It is not known to be false
3) It was published or made publicly before 2006.
4) It is Important:
(It is based on some appealing heuristic or idea;
refuting it will be important etc.)
5) IT IS NOT just the negation of a famous commonly believed conjecture.
As always with big list problems please make one conjecture per answer. (I am not sure this is really a big list question, since I am not aware of many such outrageous conjectures. I am aware of one wonderful example that I hope to post as an answer in a couple of weeks.)
Very important examples where the conjecture was  believed as false when it was made but this is no longer the consensus may also qualify!
Shmuel Weinberger described various types of mathematical conjectures. And the type of conjectures the question proposes to collect is of the kind:

On other times, I have conjectured to lay down the gauntlet:  “See,
you can’t even disprove this ridiculous idea."

Summary of answers (updated:  March, 13, 2017  February 27, 2020):

*

*Berkeley Cardinals exist


*There are at least as many primes between $2$ to $n+1$ as there are between $k$ to $n+k-1$


*P=NP


*A super exact (too good to be true) estimate for the number of twin primes below $n$.


*Peano Arithmetic is inconsistent.


*The set of prime differences has intermediate Turing degree.


*Vopěnka's principle.


*Siegel zeros exist.


*All rationally connected varieties are unirational.


*Hall's original conjecture (number theory).


*Siegel's disk exists.


*The telescope conjecture in homotopy theory.


*Tarski's monster do not exist (settled by Olshanski)


*All zeros of the Riemann zeta functions have rational imaginary part.


*The Lusternik-Schnirelmann category of $Sp(n)$ equals $2n-1$.


*The finitistic dimension conjecture for finite dimensional algebras.


*The implicit graph conjecture  (graph theory, theory of computing)


*$e+\pi$ is rational.


*Zeeman's collapsing conjecture.


*All groups are sofic.
(From comments, incomplete list) 21. The Jacobian conjecture; 22. The Berman–Hartmanis conjecture 23. The Casas-Alvero conjecture 24. An implausible embedding into $L$ (set theory). 25. There is a gap of at most $\log n$ between threshold and expectation threshold (Update: a slightly weaker version of this conjecture was proved by Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Jinyoung Park!; Further update: the conjecture was fully proved by Jinyoung Park and Huy Tuan Pham ). 26. NEXP-complete problems are solvable by logarithmic depth, polynomial-size circuits consisting entirely of mod 6 gates. 27. Fermat had a marvelous proof for Fermat's last theorem. (History of mathematics).
 A: $P=NP$
Let me tick the list:

*

*Most likely false, because, as Scott Aaronson said "If $P = NP$, then the world would be a profoundly different place than we usually assume it to be."


*Yes, it's The Open Problem in computational complexity theory


*Yes, it's old


*It's important, again quoting Scott: "[because if it were true], there would be no special value in "creative leaps," no fundamental gap between solving a problem and recognizing the solution once it's found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss..."


*It's an equality rather than a negation
A: The so-called Lovász conjecture states:

Every finite connected vertex-transitive graph contains a Hamiltonian path.

One reason I think that this conjecture fits what you're asking for is that, even though it is standardly called a conjecture, it is not clear how many people believe it. László Lovász himself originally phrased the question negatively, inviting the reader to find a counterexample. László Babai has vocally argued that there is no reason to believe it is true, and that the value of the question lies in how it highlights our poor understanding of obstructions to Hamiltonicity.  For further information, see Hamiltonian paths in Cayley graphs by Pak and Radoičić.
A: I've heard that Roger Heath-Brown has presented the following "conjecture" at several conferences, most likely to illustrate our poor understanding of the topic more than because he actually believes it to be true.
Let $\pi_2(x)$ denote the number of twin primes less than $x$. Then


$\pi_2(x) = c \int_{0}^{x}\frac{dt}{\log^2 t} + O(1)$


where $c$ is the twin prime constant. 
In other words, the twin prime asymptotic holds with error term is $O(1)$.
A: I propose Edward Nelson's "conjecture" that Peano's arithmetic is inconsistent. 
First, to be honest, I am not aware that he stated it as "conjecture", using that word, but this is something he said he believed to be true, and that he wasn't able to prove (except for a little while but a mistake was discovered by Terry Tao and others independently) though he tried a lot. So a conjecture it is, in the sense this word currently has. 
It is also certainly "outrageous", in the usual sense of the word -- to many mathematicians, according to my own experience, the simple mention of it provokes disbelief, sarcasm, sometimes outright hostility. 
But let's check  that it is also "outrageous" in the sense of this question. 1) It is most certainly false, or at least this is what most mathematicians, including myself, think. 2) But it is certainly not known to be false -- not now, not ever. 3) Nelson made his program public much before 2006. 4) it is obviously extremely important. 5) The negation, that is the assertion that "Peano's arithmetic is consistent" was once a conjecture by Hilbert, but since Gödel it cannot be called a conjecture anymore, since we know it cannot be proven (in a system etc.)
Let me add that it also satisfies something Gil Kalai added in comment, namely "I prefer examples where [...] the proposer proposes the conjecture genuinely suggesting that it is true".
A: From this Math Overflow question, Joel David Hamkins wrote:

I once heard Harvey Friedman suggest that the set of prime-differences, that is, the set of all natural numbers $n$ for which there are primes $p,q$ with $p-q=n$, as a possible candidate for all we knew for an intermediate Turing degree — a noncomputable set between $0$ and $0'$ — that was natural, not specifically constructed to have that feature.

I've also heard others (albeit more recently than 2006) conjecture that Hilbert's 10th problem for rationals is an intermediate degree. 
Really, any conjecture that there is a natural intermediate degree is outrageous (although not exactly formal enough to refute). 
A: Vopěnka's Principle
It fits here perfectly except that it has never been called a conjecture. Vopěnka himself was convinced it was wrong! But I will better just post a section from page 279 of Adámek and Rosický ``Locally presentable and accessible categories.'':

The story of Vopěnka's principle (as related to the authors by Petr Vopěnka)
  is that of a practical joke which misfired: In the 1960's P. Vopěnka was repelled by the multitude of large cardinals which emerged in set theory.
  When he constructed, in collaboration with Z. Hedrlín and A. Pultr,
  a rigid graph on every set (see Lemma 2.64), he came to the conclusion
  that, with some more effort, a large rigid class of graphs must surely be
  also constructible. He then decided to tease set-theorists: he introduced
  a new principle (known today as Vopěnka's principle), and proved some
  consequences concerning large cardinals. He hoped that some set-theorists
  would continue this line of research (which they did) until somebody showed
  that the principle is nonsense. However, the latter never materialized - after
  a number of unsuccessful attempts at constructing a large rigid class
  of graphs, Vopěnka's principle received its name from Vopěnka's disciples.
  One of them, T. J. Jech, made Vopěnka's principle widely known. Later
  the consistency of this principle was derived from the existence of huge
  cardinals: see [Powell 1972]; our account (in the Appendix) is taken from
  [Jech 1978]. Thus, today this principle has a firm position in the theory of
  large cardinals. Petr Vopěnka himself never published anything related to
  that principle.

A: This one is due to Errett Bishop: "all meaningful mathematics is reducible to finite calculations with strings of $0$s and $1$s" (imho Bishop formulated this not as a conjecture but as an article of faith but that doesn't necessarily affect the truth or falsity thereof).
A reference for Bishop's claim is his article "Crisis in contemporary mathematics" (the link is to the mathscinet review of the article) which discusses the constructivist opposition to a principle called LPO ("limited principle of omniscience") related to the law of excluded middle. The LPO is discussed starting on page 511 of the article.
A: I don't know about "ridiculous", but there is Hall's Conjecture in its original form:

There is a positive constant $C$ such that for any two integers $a$ and $b$ with $a^2 \neq b^3$, one has $$|a^2-b^3|> C \sqrt{|b|}\;.$$

A: Existence of Siegel zeros.
1) If we are to believe (like most mathematicians do) in the generalized Riemann hypothesis, this is completely false. I wouldn't necessarily call this ridiculous or outrageous, but within the evidence we have it is rather unlikely to hold.
2) Nonexistence of Siegel zeros is a problem wide, wide open, nowhere near close to being resolved.
3) According to the Wikipedia article, this type of zeros was considered back in 1930s, and earlier by Landau, but I don't know if they have explicitly stated the conjecture. GRH was posed back in 1884 though.
4) They are immensely useful in many applications, since if they exist, primes in certain arithmetic progressions "conspire" to have certain non-uniform distribution. I'm no expert, but here some uses are listed (see also this blog post by Terry Tao).
5) It implies the negation of GRH, but the negation of a statement itself is quite an awkward statement, saying "yeah, zeros might exist, but not too close to $1$".
A: The "conjecture" in algebraic geometry that all rationally connected varieties are unirational comes to mind. It's usually thrown around as a way of saying, "See, we know so little about what varieties can be unirational that we can't prove a single rationally connected variety isn't." Unirationality implies rational connectedness, but I think almost everyone believes the converse should be false.
Some background: Algebraic geometers have been interested for a long time in proving that certain varieties are or are not rational (very roughly, figuring out which systems of polynomial equations can have their solutions parametrized.) Clemens and Griffiths showed in 1972 that a cubic hypersurface in $\mathbb{P}^4$ is irrational. Since then, there's been a lot of progress in rationality obstructions e.g., Artin-Mumford's obstruction via torsion in $H^3$, Iskovskikh-Manin on quartic threefolds, Kollar's work on rationality of hypersurfaces, and most recently, Voisin's new decomposition of the diagonal invariants which have led to major breakthroughs. 
On the other hand, unirationality has proved a far harder notion to control, and to my mind the biggest open question in this area is to find any obstruction to unirationality.
A: I'm surprised no one has mentioned it, but the first one that comes to my mind is this.

$e+\pi$ is rational.

I think most mathematicians would agree that it is ridiculous. It would follow from Schanuel's conjecture that it is false, but as far as I know, the conjecture is wide open, and when it comes to (ir)rationality of $e+\pi$, more or less all that is known is the (elementary) fact that either $e+\pi$ or $e\cdot\pi$ is transcendental (naturally, we expect both of them to be transcendental, so it doesn't really get us any closer to a proof).
I'm not sure when it was made publicly, but it is very natural and unlikely to not have been considered before (I heard about it as an undergrad around 2010). I think it is quite important in that it is an obvious test case for Schanuel's conjecture, and in that it would certainly be quite shocking if it was true.
(Caveat: I am not a specialist, so if someone more competent can contradict me, please do!)
A: I think the Erdős-Szekeres conjecture (see the Wikipedia article on the Happy Ending Problem), which says that the smallest $N$ for which any $N$ points in general position in the plane contain the vertices of some convex $n$-gon is $N=2^{n-2}+1$, is a little bit ridiculous in that there is not a lot of reason to believe it. Indeed, only the cases $n=3,4,5,6$ are known, and while it is known that $2^{n-2}+1$ is a lower bound for this $N$, I really am not aware of any other substantial evidence in favor of this conjecture other than that it's a very nice pattern.
A: I was giving a talk several years ago about the conjectured linear independence (over $\Bbb Q$) of the ordinates of the zeros of the Riemann zeta function, and Lior Silberman crystallized our current lack of knowledge into a "Refute this!" statement:

If $\zeta(x+iy)=0$, then $y\in\Bbb Q$.

(In other words, even though the imaginary parts of the nontrivial zeros of $\zeta(s)$ are believed to be transcendental, algebraically independent, and generally unrelated to any other constants we've ever seen ... we currently can't even prove that a single one of those imaginary parts is irrational!)
This "conjecture" can be extended to Dirichlet $L$-functions, and perhaps even further (though one needs to be careful that we don't allow a class of $L$-functions that includes arbitrary vertical shifts of some of its members).
A: For a prime $p$, an infinite group $G$ is a Tarski monster if each of its proper subgroups has order $p$.
If I am correctly informed, then the Tarski monster was defined to demonstrate our poor understanding of infinite groups, because such monsters obviously don't exist, it should be easy to prove that they don't exist, but we cannot prove it.
Then Olshanskii proved that Tarski monsters do exist for all large primes, and by now many people believe that "large" means something like $p\geq 11$.
A: Here is my favorite.
Spherical simplex is called rational, if all its dihedral angles are rational multiples of $\pi.$
Rational Volume Conjecture:
Volume of every rational spherical simplex  is a rational multiple of $\pi^2.$
It is  true in all known examples, but is expected to be false. The conjecture first appeared in  "Differential characters and geometric invariants" paper of Cheeger and Simons, p.73. (Dis)proving this conjecture would lead to interesting results in scissor congruence theory and group homology.
A: Seen this, I believe really striking, example yesterday here on MO: although the question Complex vector bundles that are not holomorphic is from 2009, a recent post by algori suggests that this is still open.
And the really ridiculous conjecture is that all topological vector bundles on $\mathbb C\mathbf P^n$ are algebraic.
The question is described as an open problem in Okonek-Schneider-Spindler (1980) but I believe must have been asked much earlier.
A: If the holomorphic map $f:\mathbb{C}\to\mathbb{C}$
has a fixed point $p$, and the derivative $\lambda := f'(p)$ equals $e^{2\pi i \theta}$ (with irrational $\theta$), one can ask if $f$ is conjugate to $z\mapsto\lambda\cdot z$ in a neighborhood of $p$. If it exists, the largest domain of conjugacy is called a 'Siegel disk'. Two properties to keep in mind are:


*

*A Siegel disk cannot contain a critical point in its interior (boundary is ok).

*The boundary of a Siegel disk belongs to the Julia set of $f$.


Quadratic maps can have Siegel disks, but not for just any $\theta$; the number theoretical properties of this 'rotation number' are relevant. However, if $\theta$ is Diophantine, the boundary of the disk is well behaved (Jordan curve, quasi-circle...)
... But the boundary of a Siegel disk can also be wild; for instance, it can be non-locally connected. The outrageous conjecture that has been floating around is that:


*

*There is a quadratic polynomial with a Siegel disk whose boundary equals the Julia set.


Since a quadratic Julia set is symmetric with respect to the critical point, a quadratic Siegel disk would have a symmetric preimage whose boundary also equals the Julia set, but the unbounded component of the complement (Fatou set) also has boundary equal to the Julia set, so our conjectured Siegel disk would form part of a 'lakes of Wada' configuration.
A: The Lusternik-Schnirelmann category of the Lie groups $Sp(n)$.  Since $Sp(1) = S^3$, $\mathrm{cat}(Sp(1)) = 1$.  In the 1960s, P. Schweitzer proved that $\mathrm{cat}(Sp(2)) = 3$.  Based on this, a folklore conjecture emerged that in general $\mathrm{cat}(Sp(n)) = 2n-1$. In 2001, it was proved that $\mathrm{cat}(Sp(3)) = 5$, so maybe it's true?  
A: $S^6$ has a complex structure.
I don´t know if this apply, but this has a nice story. It has been "published" to to be true and now Atiyah has a short paper on arxiv claiming to be false, other important mathematicians has also work on this problem. According to LeBrun this would be a minor disaster. 
A: Let us say a graph class $\mathcal{C}$ is small if it has at most $n^{O(n)}$ graphs on $n$ vertices. The implicit graph conjecture states that every small, hereditary graph class has an adjacency labeling scheme (ALS) with a label decoder that can be computed in polynomial time (a formal definition of ALS is given at the end of the answer).
Initially, this was posed as question by Kannan, Naor and Rudich in their paper Implicit Representation of Graphs which appeared at STOC '88. It was restated as conjecture by Spinrad in the book Efficient Graph Representations (2003). 
It is important because it would imply the existence of space-efficient representations for all small, hereditary graph classes where querying an edge requires only polylogarithmic time with respect to the number of vertices of the graph. 
As far as I know there is no consensus about whether this conjecture should be true or not. However, from my perspective it would be an immense surprise if it holds for the following reason. The concept of adjacency labeling schemes can be defined with respect to arbitrary complexity classes. For a complexity class $\text{C}$ (more formally, a set of languages over the binary alphabet) we can define the class of graph classes $\text{GC}$ as the set of all graph classes that have an ALS with a label decoder that can be computed in $\text{C}$. It can be shown that $\text{G1EXP} \subsetneq \text{G2EXP} \subsetneq \text{G3EXP} \dots \subsetneq \text{GR} \subsetneq \text{GALL}$ where $\text{kEXP}$ is the set of languages that can be computed in time $\exp^k(\text{poly}(n))$, $\text{R}$ is the set of all decidable languages and $\text{ALL}$ is the set of all languages. I find it hard to believe that every small, hereditary graph class falls down through all these classes and just happens to sit in $\text{GP}$ (the choice just seems too arbitrary and weak). In fact, there are natural graph classes such as disk graph or line segment graphs for which it is not even known whether they are in $\text{GALL}$. Additionally, a graph class $\mathcal{C}$ is in $\text{GALL}$ iff $\mathcal{C}$ has a polynomial universal graph, i.e. a family of graphs $(G_n)_{n \in \mathbb{N}}$ such that $|V(G_n)|$ is polynomially bounded and $G_n$ contains all graphs from $\mathcal{C}$ on $n$ vertices as induced subgraph. It already seems doubtful to me that every small, hereditary graph class has such polynomial universal graphs. 
An ALS is a tuple $S=(F,c)$ where $F \subseteq \{0,1\}^* \times \{0,1\}^*$ is called label decoder and $c \in \mathbb{N}$ is the label length. A graph $G$ with $n$ vertices is represented by $S$ if there exists a labeling $\ell \colon V(G) \rightarrow \{0,1\}^{c \lceil \log n \rceil}$ such that for all $u,v \in V(G)$ it holds that $(u,v) \in E(G) \Leftrightarrow (\ell(u),\ell(v)) \in F$. A graph class $\mathcal{C}$ has an ALS $S$ if every graph in $\mathcal{C}$ can be represented by $S$ (but not necessarily every graph represented by $S$ must be in $\mathcal{C}$). 
A: Let me promote a comment of Terry Tao into an answer and mention two conjectures.
(See, e.g., this paper and this paper for more information.)
Conjecture 1: (a question by Gromov) Every group is sofic.
Conjecture 2: (a question by Connes): Every group is hyperlinear
Conjecture 1 and 2 as well as several other problems can be asked in the context of approximate homomorphisms of a group $\Gamma$. If the target groups are symmetric groups, and the approximation is in terms of normalized Hamming norm then this leads to Conjecture 1. If the target groups are $U(n)$ and the notion of approximation is in terms of the normalized Hilbert Schmidt norm then this leads to Conjecture 2. A counter example to Conjecture 2 leads also to a counter example to Connes 1976 embedding conjecture that was refuted a few days ago by Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen in their paper MIP*=RE, which gives us additional reasons to consider Conjecture 1 and 2 as outragous conjectures!  
A: A long-standing conjecture in Number Theory is that for each positive integer $n$ there is no stretch of $n$ consecutive integers containing more primes than the stretch from 2 to $n+1$. Just looking at a table of primes and seeing how they thin out is enough to make the conjecture plausible. 
But Hensley and Richards (Primes in intervals, Acta Arith 25 (1973/74) 375-391, MR0396440) proved that this conjecture is incompatible with an equally long-standing conjecture, the prime $k$-tuples conjecture. 
The current consensus, I believe, is that prime $k$-tuples is true, while the first conjecture is false (but not proved to be false). 
A: W. Hugh Woodin, at a 1992 seminar in Berkeley at which I was present, proposed a new and ridiculously strong large cardinal concept, now called the Berkeley cardinals, and challenged the seminar audience to refute their existence. 
He ridiculed the cardinals as overly strong, stronger than Reinhardt cardinals, and proposed them in a "Refute this!" manner that seems to be in exactly the spirit of your question. 
Meanwhile, no-one has yet succeeded in refuting the Berkeley cardinals. 
A: The finitistic dimension conjecture for finite dimensional algebras states that the supremum of all projective dimensions of modules having finite projective dimension is finite.
It it just proven for some very special classes of algebras and in general there seems to be no reason why this should be true.
References: https://arxiv.org/pdf/1407.2383v1.pdf
http://www.math.uni-bonn.de/people/schroer/fd-problems-files/FD-FinitisticDimConj.pdf
A: It seems to me that Zeeman's collapsing conjecture satisfies the criteria given. The Zeeman conjecture implies both the Poincaré conjecture (proved in 2003) and the Andrews-Curtis conjecture.
The following is a quote from Matveev's book, where it is proved that ZC restricted to special polyhedra is equivalent to the union of PC and AC.

Theorem 1.3.58 may cast a doubt on the widespread belief that ZC is
  false. If a counterexample indeed exists, then either it has a “bad”
  local structure (is not a special polyhedron) or it is a
  counterexample to either AC or PC.

