Density of rigid compatible metrics

Determine whether the set \(R(X)\) of compatible metrics on a metrizable space \(X\) with trivial bijective self-isometry group is dense in \((Met(X),D_X)\) whenever \(|X|\ge3\).

Background

A compatible metric is rigid when its only bijective self-isometry is the identity. The paper proves density of rigid metrics in several settings, including compact metrizable spaces with at least three points and totally bounded compatible metrics on arbitrary metrizable spaces with at least three points, as well as certain strongly zero-dimensional spaces.

The unresolved issue is density in the uniform topology for an arbitrary metrizable space and an arbitrary compatible target metric. The paper specifically notes that proper metrics need not be dense in the full space of compatible metrics, so the proper-metric approximation theorem does not settle this question.