The quiver-variety conjecture for framed sheaves on root stacks

Let $X_k$ be the ALE space, let ${\mathscr X}_k$ be its associated root stack with divisor ${\mathscr D}_\infty$, and let ${\mathcal F}_\infty^{0,\vec w}$ be the framing data indexed by $\vec w$. For rank $r$, discrete data $\vec u$ and discriminant $\Delta$, write ${\mathcal M}_{r,\vec u,\Delta}({\mathscr X}_k,{mathscr D}_\infty,{\mathcal F}_\infty^{0,\vec w})$ for the corresponding moduli space. Let ${\mathcal M}_{\xi_k}(\vec s,\vec w)$ be the Nakajima quiver variety with stability parameter $\xi_k$, where $\vec s\in\mathbb N^k$. Quiver-variety conjecture. For a suitable choice of $\vec s\in\mathbb N^k$,

$$\mathcal M_{r,\vec u,\Delta}({\mathscr X}_k,{mathscr D}_\infty,{\mathcal F}_\infty^{0,\vec w})\cong {\mathcal M}_{\xi_k}(\vec s,\vec w).$$

The conjecture would identify these moduli spaces of framed sheaves with Nakajima quiver varieties and thereby relate their geometry to the instanton moduli spaces on $X_k$. The cited result establishes that the instanton moduli space occurs as an open subset, but the stated isomorphism is not established in the supplied text.