Why are the quantum measurements the focus of this paper? The real source of entropy is the random bits being generated to decide which measurement to perform; the measurement actually being performed is just an after-show.

Regardless, I think it's clear that assumption `(i)` is the one that breaks: the system affects the decision of what to measure. The stream of random bits will consume all the neg-entropy in the system, cause heat death, and thereby break whatever mechanism was doing the measuring (e.g. if it was you, then you would die).