Can quantum hardness be demonstrated?
"Classical hardness" is demonstrated in the very same way, namely, we find a problem that is believed to be hard then we reduce the task of breaking the security of the scheme to the task of solving that problem.

So, it is kind of the same situation, because maybe there exist classical polynomial-time solutions to the problem used and those solutions simply were not discovered yet.

The main difference is that, in general, we are far more used to classical computers than to quantum ones and maybe saying that a problem is believed to be classical hard is more meaningful than the quantum analogous because we have really studied those problems a lot in the classical setting...