So, a proposed test:

That test only works if you take quantum measure as probability in the first place.

Are you certain of that? In what other way do you interpret measure that produces a different anticipated experience in this situation? Is there a good article that explains this topic?

Unless I'm missing something, it doesn't matter whether we take measure as probability or not, there will be an asymmetry in the measure between the experiment performed and experiment not performed pathways, when there would not be in the normal quantum coin case. Or are you saying that while the quantum measure is different in the different pathways, we have no way to measure it? If so, then what do you actually mean by quantum measure, given that we can't measure it? (Or is there some other way to measure it, that somehow can't be turned into a similar experimental test?) And, if we can't measure it or any effects from it, why do we believe it to be "real"? What causal pathway could possibly connect to our beliefs about it?