The wreath-product Hilbert quotient morphism conjecture

Let $\Gamma$ be a finite subgroup of $SL_2(\mathbb{C})$, let $\Gamma_n=\Gamma^n\rtimes S_n$, and let $X_{\Gamma,n}$ be the union of connected components of the $\Gamma$-fixed locus in $(\mathbb{C}^2)^{[n|\Gamma|]}$ consisting of $\Gamma$-invariant ideals $I$ for which $\mathbb{C}[x,y]/I$ is isomorphic to $R^n$ as a $\Gamma$-module, where $R$ is the regular representation of $\Gamma$. Let $\varphi:\mathbb{C}^{2n}//\Gamma_n\to X_{\Gamma,n}$ be the natural morphism induced by the universal family. The wreath-product Hilbert quotient morphism conjecture. The morphism $\varphi$ is an isomorphism.

This identifies the Hilbert quotient with the smooth variety $X_{\Gamma,n}$ and would provide a geometric realization of the wreath-product side of the McKay correspondence; the source supplies the construction of the morphism but no resolution status.