The reason why the Everett Interpretation is needed for the experiment to be valid is that according to it decoherence results is a superposition of quantum mechanical states, and not a collapse. If you believe in another interpretation, there has been a collapse, and so the experiment demonstrated nothing.
But we are bringing gravity into the mix. Gravity works differently than the other forces. Do space-time structures also exist in a superposition of states? Or does QM work against an essentially classical gravitational space-time background?
Without a theory combining quantum mechanics and gravity, the answer is not obvious. The fact that the superpositions do NOT interfere gravitationally means that either there was no quantum mechanical superposition (collapse actually happened) or that gravity can also exist in superimposed states. If you believe in the Everett interpretation, then there must have been a quantum mechanical superposition, and this tells us something non-trivial about a quantum theory of gravity.