Mirror-symmetry correspondence for punctured class-$S$ theories

Let $V_{g,n}$ be the vertex operator algebra associated with a genus-$g$ class-$S$ theory with $n>0$ regular punctures, and let $\mathcal{M}_{g,n}^T=\{M_a\}$ be the fixed manifolds of the corresponding moduli space. Assume $0<\alpha_n<\cdots<\alpha_1<1/4$, where $\alpha_i$ are the parabolic parameters. Let $h(L_a)$ be the conformal dimension of the highest-weight state of the simple module $L_a$, let $\omega^{(i)}$ be the fundamental weight of the $i$th flavor $\mathfrak{su}(2)$, and let $[\lambda_1,\ldots,\lambda_n]$ denote its flavor representation. Write $\mu_\text{max}$ for the maximal moment-map value.

Mirror-symmetry conjecture. When $n>0$, there is a bijection between the simple modules $\{L_a\}$ of $V_{g,n}$ and the fixed manifolds $\mathcal{M}_{g,n}^T=\{M_a\}$ such that, for even $n$,

$$\left.\mu_{M_a}-\mu_\text{max}\right|_{\alpha_i\mapsto-\omega^{(i)}}=h(L_a)+\sum_{i=1}^n\lambda_i\omega^{(i)},$$

and, for odd $n$,

$$\left.\mu_{M_a}-\mu_\text{max}\right|_{\alpha_1\mapsto\frac{1}{2}+\omega^{(1)},\,\alpha_{i>1}\mapsto\frac{1}{2}-\omega^{(i)}}=h(L_a)+\sum_{i=1}^n\lambda_i\omega^{(i)}.$$

Moreover, the Jordan type of modular $T$ (respectively, $STS$) is

$$\left[(1-\delta_{\mu(M_a),0})(\dim M_a+1)+g\right]_{M_a\in\mathcal{M}_{g,n}^T}$$

for even (respectively, odd) $n$.

This conjecture extends the proposed correspondence between VOA modules and fixed manifolds in the presence of puncture flavor symmetries. The source does not state a resolution, so it remains open.