The Hilbert-Chow and symmetric-power diagram for Kleinian quiver varieties

Let $0 \in J \subseteq I$ and set $K \coloneqq I \setminus J$. Let $X_J$ be the corresponding partial resolution of the Kleinian singularity, let $g:X_J\to X_0$ be the natural morphism, and let $\sigma_K$ be the associated stability cone. The Hilbert-Chow and symmetric-power conjecture. If $\theta$ lies in the relative interior of $\sigma_K$ and $\theta'$ lies in the relative interior of $\sigma_K \cap \delta^{\perp}$, then there is a commutative diagram

\begin{tikzcd} \operatorname{Hilb}^n(X_J) \arrow[r, "\sim"] \arrow[d] & \operatorname{\mathfrak{M}}_{\theta}(n \delta,w) \arrow[d] \\\\ \operatorname{Sym}^n(X_J) \arrow[r, "\sim"] \arrow[d] & \operatorname{\mathfrak{M}}_{\theta'}(n \delta,w) \arrow[d] \\\\ \operatorname{Sym}^n(X_0) \arrow[r, "\sim"] & \operatorname{\mathfrak{M}}_0(n \delta,w), \end{tikzcd}

in which all morphisms are birational and projective, the horizontal morphisms are isomorphisms, the upper and middle horizontal morphisms are induced by $g_*$, the upper left vertical morphism is the Hilbert-Chow morphism, the lower left vertical morphism is $\operatorname{Sym}^n(g)$, and the right vertical morphisms arise from GIT specialisation. The conjecture identifies the remaining intermediate partial resolutions in the diagram of Nakajima quiver varieties.