In fact, the stimulus to Copenhagen interpretation was something like "we only nowknow the things we get from measurements, so we don't care what happens in between". Consider two statements:
A. We get this state with this probability upon doing measurements and we don't care what happens in between.
B. Between measurements system is at all possible states simultaneously.
If you want to falsify Copenhagen interpretation, you have to falsify statement A, not statement B.
Ninja edit: Some authors claim (I don't know where do they know it from) that when Maxwell was working on his theory of electrodynamics, he was imagining the space filled with little gears and cranks which transfer the influences. However, when you want to falsify Maxwell, you aim at his predictions (what will we get if we measure this thing in that way) not at those gears.