Most 'obvious' open problems in complexity theory What open problems in computational complexity theory have the most 'obvious' answers, regardless of whether that answer is true or false?  The problems I'm talking about certainly have more 'obvious' answers than P =?= NP.
I'll start this with
ZPP =?= EXPTIME
 A: $AC^0[6] vs. NP$
In other words, it is open whether $SAT$ has polynomial size constant depth circuits using gates $\land$, $\lor$, $\lnot$, $mod_6$. 
I think it is more than obvious, it is kind of embarrassing that we can't prove they are not equal. Note that we know $AC^0[p]$ cannot compute $mod_q$ for $p\neq q \in Primes$. I think the fact that we can't prove a similar result for $mod_6$ is just lack of tools, the tools we have break as soon as we have two $mod$ gates.
Edit
Note that the barrier results does not seem to apply here.
Uniformity condition: language of direct connection graphs is in $DLogTime$.
A: $BQP\subseteq ?PH$
We know that simulating quantum mechanics requires polynomial space, but still is open if wether there are problems that only quantum computers can solve efficiently.
A: Is integer factorization outside of P?
A: GI $\in$ P. We know that there are bad consequences if GI is NP-complete, 
p.s GI is graph isomorphism
A: A) Multiplication (of $n$--bit numbers) obviously takes longer than addition.
B) Sorting $n$ objects obviously takes longer than linear time.  (I suppose that this is obvious because it is true: there is a lower bound of $n \log n$ for comparison-based sorts.  However, radix sort of integers can be faster.  In fact, for any collection (of a particular type of object), there just might be a super clever way to sort it faster than $n \log n$....)
A: In 1990, my intuition (and I don't think it was just mine) was that IP couldn't possibly contain PSPACE. Intuition was wrong.
A: The following two statements are really "obviously false", but are still open:
$EXP^{NP} \subseteq$ depth-2-$TC^0$
$EXP^{NP} \subseteq$ depth-2-$AC^0[6]$
Just as a reminder:  


*

*$EXP^{NP}$ is exponential time plus an oracle for NP.  It contains $NEXP$ (nondeterministic exponential time), $EXP$, and $NP$.

*By "depth-2-$TC^0$" I mean the class of polynomial-size, depth-two circuit families where each gate is an arbitrary threshold function -- i.e., if it has $m$ Boolean inputs $x_1, \dots, x_m$, it is defined by reals $a_1, \dots, a_n, \theta$ and has output 1 iff $\sum a_i x_i \geq \theta$.

*By depth-2-$AC^0[6]$ I mean the class of polynomial-size, depth-two circuit families where each gate is a "standard" $MOD_6$ gate: if it has $m$ Boolean inputs $x_1, \dots, x_m$, it has output 1 iff $\sum x_i \neq 0$ mod 6.
Expanding on the second open problem:  Given $A \subseteq \mathbb{Z}_6$, define an $A$-$MOD_6$ gate to be one which outputs 1 iff $\sum x_i \in A$ mod 6.  
The most embarrassing open problem in circuit complexity may be the following:  Show that for all possible subsets $A$, the AND function requires superpolynomial-size depth-2 circuits of $A$-$MOD_6$ gates.  
[PS: Thanks to Arkadev Chattopadhyay for explaining some of these $MOD_6$ problems to me.]
