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.