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):
Hall's original conjecture (number theory).
Tarski's monster do not exist (settled by Olshanski)
The implicit graph conjecture (graph theory, theory of computing)
(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!). 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).