There are a two cases:
BQP contains something BPP doesn't: Then the thesis is false as long as we can realize a quantum computer, as then there exists something which can be solved in polynomial time (with bounded error) on a machine in the universe which cannot be done on a classic Turing machine as efficiently.
BQP is a subset of BPP (equal or ~~strict~~):
Then the state of the thesis is unknown. There is still some chance that is is false, either by a superpolyomial speedup somewhere higher up the hierarchy (not too sure about this one, my complexity theory is not sharp enough to rule out that this is impossible given BQP is a subset of BPP), or that there exists another realizable model of computation other than quantum computing that beats the randomized turing machine.
EDIT:
It appears that BPP (trivially) a subset of BQP, so that the strict inclusion of BQP in BPP is impossible, though BQP = BPP is possible.
BQP is a subset of BPP (equal or ~~strict~~): Then the state of the thesis is unknown. There is still some chance that is is false, either by a superpolyomial speedup somewhere higher up the hierarchy (not too sure about this one, my complexity theory is not sharp enough to rule out that this is impossible given BQP is a subset of BPP), or that there exists another realizable model of computation other than quantum computing that beats the randomized turing machine.
EDIT: It appears that BPP (trivially) a subset of BQP, so that the strict inclusion of BQP in BPP is impossible, though BQP = BPP is possible.