OK, the example linked is defective, in that there are two different operations that get the same result when the machine reverses its x-axis measurement. The first is the time-reversal of the measurement operation; the second is the recreation of the state created by measurement 1. You seem to be saying that the Copenhagenite would assume the latter.

Here is a modification of the experiment that tests the idea of collapse more severely. Instead of preparing an electron in a |+z> or |-z> state, I prepare an entangled pair of electrons with opposite z-axis spin (a spin anti-correlated pair). I now give one electron to the machine intelligence, which measures its spin in the x-axis, and then applies the time-reversal of the measurement, restoring the electron's original state and erasing its memory of the x-axis state. It then passes the electron back to me, and I measure the two electrons' z-axis spins.

If the machine intelligence's measurement had caused a collapse, the anti-correlation would be erased. But in fact everything we know about quantum mechanics says that the electrons should remain anti-correlated.