Locally classical mediators with non-locally tomographic couplings can generate entanglement, so the BMV experiment does not by itself prove spacetime is quantum.
On the Role of Locality in the Bose-Marletto-Vedral Effect
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Two of us recently proposed an entanglement-based witness of non-classicality, which can be applied to testing quantum effects in gravity in what is known as the Bose-Marletto-Vedral (BMV) effect. The witness is based on this idea: if a system can create entanglement between two quantum probes by local means only, then it must be non-classical. In this note we discuss the role of locality as an assumption for the theorem supporting the witness; we also discuss other related notions of locality and comment on their mutual relations.
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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.