The noncompact extension of the orbifold Hecke algebra isomorphism

Let $X$ be a noncompact complex manifold equipped with an action of a finite subgroup $G\subset\operatorname{Aut}(X)$. Theorem~ gives an isomorphism in the compact case.

Noncompact extension conjecture. The isomorphism of Theorem~ also holds for such noncompact $X$ and $G$.

This conjecture proposes extending the theorem from compact to noncompact complex manifolds; the supplied text gives no resolution or further conditions.