What would show a human mind is/is not reducible to a Turing machine?
To answer your question:


  Is there any kind of test within our current capabilities where we can prove/disprove this assumption?


The Turing test was conceived as a way to test a particular special case of this assumption: it gives us a way to test whether a particular AI system is successful at behaving indistinguishably from a human.  Thus, this would give a plausible way to prove the assumption -- if we could come up with an AI system that is good enough to pass the Turing test.  Unfortunately, that's something we haven't been able to do, yet.