Borel complexity of the rigidity locus

Determine whether the set \(R(X)\) of compatible metrics on a metrizable space \(X\) with trivial bijective self-isometry group is a Borel subset of \((Met(X),D_X)\).

Background

The paper establishes that rigid proper metrics form a $G_\delta$ subset within the proper compatible metrics on a locally compact Polish space. It also obtains a dense $G_\delta$ subset of rigid metrics in several more specialized settings.

Those results do not determine the descriptive-set-theoretic complexity of the full rigidity locus $R(X)$ in the uniform metric space $(Met(X),D_X)$. In particular, the presence of dense $G_\delta$ subsets does not imply that the entire set $R(X)$ is Borel.