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

There is no shortage of open problems in mathematics. While a formal proof for any of them remains elusive, with the "yes/no" questions among them mathematicians are typically not working in both directions but rather have a pretty clear idea of what the answer should be. Famous conjectures such as Riemann and Collatz are supported by some very convincing heuristics, leading mathematicians to believe in their validity so strongly that they write papers based on the assumption that they are true. For other wide open problems such as $P$ vs. $NP$, one side ($P=NP$ in this case) is usually considered so unlikely to be true that almost nobody seriously works on it. Of course, whenever a "conjecture" is attached to an open question that already implies that one answer is preferred over the other – people don't conjecture $A$ and $\neg A$ simultaneously.

Are there any open mathematical questions with a yes/no answer for which we have no good reason to assume one or the other, for which we really have absolutely no idea what the true answer might be?

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.