I read the question rather differently: can any amount of past and present evidence falsify a claim about the future, insofar as it still remains the future? Of course, past and present evidence can give us ample reason to doubt certain claims that might be made about the future: but could it ever demonstratively disprove such claims? I'm not at all sure that it could. An instance of an F that isn't G can falsify the proposition that all Fs are currently G, but it can't similarly falsify the proposition that all future Fs will be G. Current evidence tells us about what is currently true or false, and to project this onto the future for the purposes of falsification is as problematic -- no more so, but also no less so -- as projecting it onto the future for the purposes of verification. And, as David Hume showed us more than 250 years ago, that there are genuine grounds for concern about the latter. The 'problem of induction' suggests that there is a certain logical circularity in any such attempt at projection, be it positive or negative.
But then, let's not forget how Karl Popper presented his doctrine of falsificationism. It was supposed to be an alternative to inductivism, one that didn't actually need to rely (at least not overtly) on any problematic assumptions concerning the uniformity of nature. So I don't think Popper would be especially concerned about the challenge that you raise. He could acknowledge that such a supposition isn't falsifiable -- and consequently would presumably conclude that it isn't scientific -- but he could then just carry on regardless. I'm not sure that, within the terms of his own programme, this would qualify as a problem at all.