The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first published in 1991 (the geometric quantization picture here and the skein theoretic approach here), and despite a tremendous amount of development since then, it is still not known whether they are equivalent! I guess that it is reasonable to say that the problem of their equivalence has been around for 20 years.

There is (at least) one important theorem, namely the asymptotic faithfulness of the mapping class group representations produced by these two quantizations, which has proofs in both settings. The two proofs are of completely different character, and are of course logically independent, since the two representations are not known to be the same (this was proved for the quantum group skein representation by Freedman, Walker, and Wang, and for the geometric quantization representation by Andersen).

Is there a good reason why the equivalence of these two viewpoints is not yet a theorem? Is there an idea for a proof, which hasn't been completed because "it's just a long calculation" or "everyone knows it's true" or "it's nice to know, but it wouldn't actually help us prove theorems"? Or is it that it's actually a hard problem that no one knows how to approach? Is it an "important" problem whose solution would have lots of consequences and applications, or at least advance our understanding of "quantization"?