Nonabelian mirror-brane conjecture for Hamiltonian group actions

Let $Y$ be a Hamiltonian $G$-manifold, with maximal torus $T$ and Weyl group $W$, and suppose the $T$-action on $Y$ extends to a Hamiltonian $G$-action. Let $\mathbb{L}^T_Y\subset C_T$ be the abelian mirror brane, and let $W\text{-}\mathrm{Hilb}_{\mathfrak{t}_\mathbb{C}}(-)$ denote the transverse Weyl-group Hilbert-scheme construction. Nonabelian mirror-brane conjecture. The subvariety $\mathbb{L}^T_Y$ is $W$-invariant; the natural morphism

$$W\text{-}\mathrm{Hilb}_{\mathfrak{t}_\mathbb{C}}(\mathbb{L}^T_Y)\to \mathbb{L}^T_Y/W$$

is an isomorphism; and the morphism

$$W\text{-}\mathrm{Hilb}_{\mathfrak{t}_\mathbb{C}}(\mathbb{L}^T_Y)\to W\text{-}\mathrm{Hilb}_{\mathfrak{t}_\mathbb{C}}(C_T)$$

factors through the image of $C_G\to W\text{-}\mathrm{Hilb}_{\mathfrak{t}_\mathbb{C}}(C_T)$. This is the proposed construction of the nonabelian mirror brane $\mathbb{L}^G_Y$ from the abelian brane; the source gives no resolution of the conjecture.