To explain quantum mechanical observations you need "collapse" (i.e. the projection postulate: immediately after a measurement the state of the quantum system is the projection on the corresponding eigenspace of the operator). I don't know what do you gain by saying that it's not "real" and that it's just "as if".

Because if we say that some of those alpha_i physically go to zero at any point (e.g., "after"), our predictions are wrong, in agreement with the paper. We have to account for the fact that those alpha_i|i> are still nonzero and "existing", and that our projection onto them is only zero for the time being. A different choice in EPR or a quantum eraser experiment may bring our projection onto those states back out of orthogonality-- or maybe not, if we never make those choices. But if we believe we have the physical freedom to manipulate our experiments, we can't get away with saying those extra "universes" (basis states) physically disappear.

In some cases it is a safe approximation to ignore those extra states for the remainder of our experiment/calculations, but with a small change to the experiment we can make that a bad approximation.

I am afraid that you have not understood the paper.

a) You do the measurement first on the "screen" side (and project the quantum state of the pair of photons according to the measurement, the "extra universes" disappear). You do then the measurements on the "idler" side (and project again the quantum state according to standard QM).

b) You modify slighly the setup to reverse the order of the measurements. You do the measurement first on the "idler" side (and project the quantum state of the pair of photons according to the measurement, the "extra universes" disappear). You do then the measurements on the "screen" side (and project again the quantum state according to standard QM).

QM predicts that the outcomes in the original experiment (a) and the "reversed" experiment (b) are the same. And those predictions are verified empirically.