You're right that part of the problem is, indeed, theoretical (it remains to be shown that BPP ≠ BQP, even if it is widely believed to be true), but part of the

You're right that part of the problem is, indeed, theoretical (it remains to be shown that BPP ≠ BQP, even if it is widely believed to be true), but part of the problem is also physical (we need to build a model of BQP---i.e. a quantum computer---to show that there is a physically realizable model of it).