This question is probably stupid since I have little information about the topic, but have we proved that it cannot be solved, or is it still an open problem?
Can you please provide me with the proof or with the recent results regarding it (if open)?
"Non-abelian" is a little broad - there are some examples of non-abelian groups having algorithms in BQP. One example is that of the Heisenberg group. If there is a lot of commutativity amongst the elements then an efficient circuit might still exist.
I'd refer to this older paper from Bacon, Childs, and van Dam; to quote the abstract:
We approach the hidden subgroup problem by performing the so-called pretty good measurement on hidden subgroup states. For various groups that can be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pretty good measurement is optimal and that its probability of success and unitary implementation are closely related to an average-case algebraic problem. By solving this problem, we find efficient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no efficient algorithm was previously known: certain metacyclic groups as well as all groups of the form $(Z_p)^r X| Z_p$ for fixed $r$ (including the Heisenberg group, $r=2$). In particular, our results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.
I'd also refer to Jordan's wonderful Quantum Algorithms Zoo, especially the entry on Non-Abelian Hidden Subgroups.
That said, there are some no-go results suggesting that general non-abelian groups do not have an efficient quantum algorithm that is of the same kind as the abelian cases; something about irreps being too close to each other (mumble-mumble don't really understand). See, e.g., Moore's very accessible lecture here.
Proving unconditional no-go results is likely out-of-bounds for now as this would split NP (where most hidden subgroup problems lie) from BQP (where efficient quantum algorithms lie), which is a very hard problem.
