I'm not convinced. They aren't actually using the semiclassical Einstein equation, they are using some simplification they call Newton Schrodinger equation. They claim that this equation can lead to distinguish a state that's exponentially close to |0> from |0>. I don't follow their whole argument.

Anyway, I wouldn't be surprised if you could actually do hard computations with semiclassical Einstein equation, because they have strong self consistency - the expectation value of the stress energy tensor curves the metric, which in turn excites the quantum vacuum and causes expectation value of the stress energy tensor. But this isn't what they use in the paper. Nobody knows if this self consistency can be achieved in all configurations, and physicists working with semiclassical gravity usually do only one iteration of the self consistency.

If someone wanted to make semiclassical gravity into quantum gravity, he'll probably assume that gravity causes measurements, which would prevent these kinds of abuses where you have superposition you're probing via gravity while keeping it intact.