The Chow-resolution conjecture for generalized punctual Hilbert schemes

Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $\operatorname{Hilb}(\mathfrak{g})$ be its generalized Hilbert scheme with Chow morphism $ch$ to $\mathfrak{h}^2/W$. Chow-resolution conjecture. There is a modified version of $\operatorname{Hilb}(\mathfrak{g})$, identifying some points, which is a smooth projective variety such that the Chow morphism is a resolution of singularities. The conjecture proposes a nonsingular projective replacement despite the fact that the original generalized Hilbert scheme is not Hausdorff; the paper gives no resolution of this claim.