One problem where that mathematicians would love to see resolved is the Smooth Poincaré Conjecture (in dimension 4): https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conj...
This is a problem where experts have no clear idea whether it holds or not (and different mathematicians have strong opinions here). For a civilized discussion on whether the Smooth Poincaré Conjecture holds, see for example https://mathoverflow.net/questions/97464/how-to-tackle-the-s...
Additionally, "by mere coincidence" :-) (which makes this problem very fitting for the drama around the recent AI proof drops), the Smooth Poincaré conjecture is a problem where Selman Akbulut, who made a lot of progress on it, posted strongly worded claims of how his results were "stolen" by other famous mathematicians in the past:
* https://selmanakbulut.com/plagiarism.pdf
* https://selmanakbulut.com/story.pdf
* https://selmanakbulut.com/story2.pdf
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Addendum: Also clearing up the situation of the Generalized Poincaré Conjecture for Diff manifolds would be very interesting:
https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conj...