The cornering and GIT-specialisation diagram for Kleinian quiver varieties

Let $0 \in J \subseteq J' \subseteq I$, set $K \coloneqq I \setminus J$ and $K' \coloneqq I \setminus J'$, and let $\sigma_{K,K'}$ be the cone defined by

$$\sigma_{K,K'} \coloneqq \left\{\theta \in \Theta_{n\delta} \\; \Bigg| \\; \begin{matrix*}[l] \theta(\delta) \geq 0 \\;, & \\ \\\\ \theta(\alpha_j) \geq (n-1)\theta(\delta) & \text{for} \\; j \in J \setminus \\{0\\} \\;, \\\\ \theta(\alpha_k) \geq 0 & \text{for} \\; k \in K \setminus K' \\;, \\\\ \theta(\alpha_k) = 0 & \text{for} \\; k \in K' \\;. \end{matrix*} \right\}.$$

Let $h_*\operatorname{\mathcal{E}}_{J'}$ and $h_*\operatorname{\mathcal{V}}_{J'}$ be the cornered algebra and module, and write $\operatorname{Quot}_{J,J'}^v$ for the corresponding Quot scheme. The cornering and GIT-specialisation conjecture. If $\theta \in C_K$, $\theta'$ lies in the relative interior of $\sigma_K$, and $\theta”$ lies in the relative interior of $\sigma_{K,K'}$, then there is a commutative diagram

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

in which all morphisms are birational and projective, the horizontal morphisms are isomorphisms induced by $g_*$, the left vertical morphisms are induced by cornering at $J'$ and $J$, respectively, and the right vertical morphisms arise from GIT specialisation. This conjectures an interpolation between the Quot scheme, the intermediate cornered Quot scheme, and the Hilbert scheme, matching the corresponding Nakajima quiver varieties.