A topometric Effros theorem

Given a continuous and isometric action of a Polish group $G$ on an adequate Polish topometric space $(X,\tau,\rho)$ and $x \in X$, we find a necessary and sufficient condition for $\overline{Gx}^\rho$ to be co-meagre; we also obtain a criterion that characterizes when such a point exists. This work completes a criterion established in earlier work of the authors.