Open mathematical questions for which we really, really have no idea what the answer is

In 4 dimensions, it is an open question as to whether there are any exotic smooth structures on the 4-sphere.

A more or less elementary example I'm quite fond of is the Erdős conjecture on arithmetic progressions, which asserts the following:

If for some set $S\subseteq \mathbb{N}$ the sum $$\sum_{s\in S}\frac{1}s$$ diverges, then $S$ contains arbitrarily long arithmetic progressions.

I've never seen a heuristic argument one way or the other - I believe the strongest known result, as of now is Szemerédi's theorem, which, more or less, states that if the lower asymptotic density of $S$ is positive (i.e. there are infinitely many $n$ such that $|[1,n]\cap S|>n\varepsilon$), then it contains arbitrarily long arithmetic progressions. There's also the Green-Tao theorem which is a special case of the conjecture, giving that the primes have arbitrarily long arithmetic progressions (and, indeed, establishes the fact for a larger class of sets as well).

Yet, neither of these suggests that the result holds in general. It's tempting to believe it's true, because it'd be such a beautiful theorem, but there's not much to support that - it's really unclear why the sum of the reciprocals diverging would have anything to do with arithmetic progressions. Still, there's no obvious examples of where it fails, so it's hard to make an argument against it either.

I believe whether or not the Thompson group $F$ is amenable is such question. The paper/article "WHAT is... Thompson's Group" mentions that at a conference devoted to the group there was a poll in which 12 said it was and 12 said it was not. There are in fact papers claiming (at least at the time) to have proofs for both sides. Here are some posts to get an idea of the "controversy": 1, 2,3.