Local structure conjecture for the compactified instanton moduli space

Let $G$ be a simple algebraic group, let $k$ be a characteristic class, and let $n,l,m$ be integers. Write ${\widehat M}_0(G,k,n)$ for the compactified moduli space of framed instantons on the blowup, $M_0(G,n')$ for the corresponding unframed compactified moduli space, and let $(\vec{k},l,m)$ denote a fixed point with instanton numbers satisfying the notation above. Let $\widetilde T$ act on these spaces, with its $T^2$ factor acting on the last two factors as specified below.

Local structure conjecture. (1) There exists a proper continuous map

$$\widehat\pi_0\colon {\widehat M}_0(G,k,n)\to M_0(G,n')$$

for some $n'$. (2) A neighborhood of the fixed point $(\vec{k},l,m)$ in ${\widehat M}_0(G,k,n)$ is isomorphic, as a $\widetilde T$-space, to a neighborhood of $(\vec{k},0,0)\times 0\times 0$ in

$${\widehat M}_0(G,k,n-l-m)\times M_0(G,l)\times M_0(G,m),$$

where the $T^2$-actions on the latter two factors are modified respectively by

$$(t_1,t_2)\mapsto (t_1,t_2/t_1),\qquad (t_1,t_2)\mapsto (t_1/t_2,t_2).$$

This conjecture supplies the proper morphism and local product model needed to analyze fixed points and equivariant intersection theory on the compactified instanton moduli space. The source does not provide evidence resolving either assertion, so their status remains open.