Has this experiment really demonstrated wave-function collapse? My question is: why did the following experiment claim that it had demonstrated the wave-function collapse?

Experimental proof of nonlocal wavefunction collapse for a single particle using homodyne measurements. M Fuwa et al. Nature Communications 6 6665 (2015), arXiv:1412.7790.

I would have no problem if they had claimed that, the experiment demonstrated the "non-local" (or: precisely quantum) steering effect. In my humble opinion, there is no logic to justify that "quantum steering effect is equivalent to the wave-function collapse". Here the wave-function collapse is defined in the strict Von-Neumann's postulate form.
I am afraid that, this type of quantum steering experiment would cause the similar misunderstanding to that caused by the Bell's type of experiments. Just a reminder that, the Bell's definition of "non-locality" has no direct relation with the Von-Neumann's wave-function collapse postulate, either. All the "relations" that people had considered/debated were based on many extra assumptions and interpretations.
Did I miss something that are really profound and important here? Many thanks!
 A: The paper doesn't explain how their predictions would differ from those of non-collapse theories. Since the paper doesn't even discuss what would be predicted without collapse, it is difficult to see how it could rule out quantum theory without collapse. Quantum theory without collapse explains all of the predictions commonly attributed to quantum theory with collapse:
http://arxiv.org/abs/1212.3245.
Variations on quantum theory that include collapse, such as the GRW theory, may or may not reproduce the predictions made in the paper, but as this is not discussed it is difficult to tell whether the results are even consistent with such a theory. As such, the title of the paper does not accurately describe its contents.
A: Yes the wave function "collapses" after Alice measures her part. I am not sure what you call the Von-Neumann collapse. Here you have a bipartite measurement with only one photon. And I guess the steering thing had never been done before with only one photon.
No there's nothing profound.
