Locally classical mediators with non-locally tomographic couplings can generate entanglement, so the BMV experiment does not by itself prove spacetime is quantum.
The Bose-Marletto-Vedral proposal in different frames of reference and the quantum nature of gravity
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Observing spatial entanglement in the Bose-Marletto-Vedral (BMV) experiment would demonstrate the existence of non-classical properties of the gravitational field. We show that the special relativistic invariance of the linear regime of general relativity implies that all the components of the gravitational potential must be non-classical. This is simply necessary in order to describe the BMV entanglement consistently across different inertial frames of reference. On the other hand, we show that the entanglement in accelerated frames could differ from that in stationary frames.
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Bose-Marletto-Vedral experiment without observable spacetime superpositions
Locally classical mediators with non-locally tomographic couplings can generate entanglement, so the BMV experiment does not by itself prove spacetime is quantum.