I'm surprised this hasn't been posted yet:

Lusztig's conjecture in modular representation theory, which would describe the characters of simple $G_1$-modules for $p$ larger than the Coxeter number, was generally believed to be true for a long time. Here $G$ is a reductive algebraic group in characteristic $p$ and $G_1$ is its Frobenius kernel. The conjecture in turn leads to character formulae (or at least, algorithms for calculating characters) for simple representations of $G$.

They were proven for large $p$ by Andersen-Jantzen-Soergel, with later contributions by others, including an explicit bound by Fiebig. But recently Geordie Williamson proved that the original condition on $p$ is not enough.

See https://mathoverflow.net/questions/138310/what-to-do-now-that-lusztigs-and-james-conjectures-have-been-shown-to-be-false for more details.