Is it proven that quantum computation is no better at solving NP complete problems than classical computation?

Is it proven that quantum computation is no better at solving NP complete problems than classical computation or it's just believed?

It is suspected that NP-complete problems cannot be solved in quantum polynomial time (i.e., that they are not in BQP), but this hasn't been proved. We don't expect a proof in the near future, since this would imply that P is different from NP.

StackExchange Computer Science Q#69233, answer score: 11