Isomorphism of quantization and Poisson deformation groupoids for all Kleinian singularities

Show that for any Kleinian singularity X, the groupoid QIso(X) of isomorphisms between quantizations U_c and the groupoid PIso(X) of isomorphisms between Poisson deformations A_c are isomorphic over the parameter space h^*/W.

Background

The survey defines two groupoids over the parameter space h^*/W: QIso(X), whose morphisms are algebra isomorphisms between quantizations U_c of X, and PIso(X), whose morphisms are Poisson isomorphisms between filtered Poisson deformations A_c of C[X]. It asks whether these two groupoids are isomorphic. Castellan proves the isomorphism for Kleinian singularities of types A and D and conjectures it for all Kleinian singularities.

This problem generalizes the Belov-Kanel–Kontsevich conjecture (proved for C^{2n}) to singular symplectic settings and connects automorphism groups of noncommutative quantizations with those of Poisson deformations.