Does condensed / pyknotic mathematics afford an (yet!) another approach to orbifold theory?

Let me admit up-front that I don’t know much about either condensed / pyknotic mathematics or about orbifold theory. But I think the following is a precise question which gets at the the above vague question.

I believe there are several approaches to defining what an orbifold is, so for safety let me restrict attention to the $(2,1)$-category $mathcal C$ of *topological* orbifolds $X$ such that $X$ is the orbifold quotient of a topological manifold $X_0$ by a properly discontinuous group action. I hope that even if there are differences between definitions of topological orbifolds, they should all at least agree on the category $mathcal C$ up to equivalence of $(2,1)$-categories.

On the other hand, every compactly generated weak Hausdorff space, and in particular every topological manifold, may be regarded as a condensed set in a canonical way. Moreover, there inclusion functor $Set to Gpd$ induces an inclusion functor $Cond(Set) to Cond(Gpd)$ from condensed sets to condensed groupoids. Let $mathcal D$ be the full, $(2,1)$-subcategory of $Cond(Gpd)$ whose objects are $(2,1)$-categorical quotients of topological manifolds by properly discontinuous group actions.

There are canonical inclusion functors $Man to mathcal C$ and $Man to mathcal D$ where $Man$ is the category of topological manifolds (where every 2-morphism is the identity).

**Question:** Is there an equivalence of $(2,1)$-categories $mathcal C simeq mathcal D$ respecting the two inclusions $Man rightrightarrows mathcal C, mathcal D$?