Namikawa-Weyl orbifold conjecture for T-sigma quiver chiral algebras

Let $(Q,\mathbf v,\mathbf w)$ be a $T^{\sigma}_{[1^n]}[\SU(n)]$ quiver, and let $\mathbb W=\mathfrak S_n$ be its Namikawa–Weyl group. Let $\mathsf D^{\mathrm{ch}}(\widetilde\CM(Q,\mathbf v,\mathbf w))$ be the associated chiral differential operator vertex algebra, and let $\Gamma$ be the embedding in the Kac–Roan–Wakimoto embedding conjecture. Namikawa-Weyl orbifold conjecture. The group $\mathbb W$ acts on $\mathsf D^{\mathrm{ch}}(\widetilde\CM(Q,\mathbf v,\mathbf w))$ by vertex-algebra automorphisms, and

$$\mathsf D^{\mathrm{ch}}(\widetilde\CM(Q,\mathbf v,\mathbf w))^{\mathbb W}\otimes \mathcal H^n$$

is the image of $\Gamma$. This predicts that the affine $\mathcal W$-algebra embedding is precisely the Namikawa–Weyl orbifold subalgebra, extending the finite $\mathcal W$-algebra invariant-theory picture to the chiral setting.