Closures of smooth orbits in generalized center constructions

Determine whether, for a closed inclusion of pro-oligomorphic groups G ⊂ H with X = H/G, the closure of a smooth G-orbit in X is a finite union of smooth G-orbits.

Background

The authors discuss a generalization of the Drinfeld center via G-equivariant sheaves on the homogeneous space X=H/G and suggest an analogue of their Yetter–Drinfeld analysis in this setting. A key structural property about orbit closures would significantly impact the description of the resulting module categories.

They pose the question explicitly as part of outlining this generalized framework.