Gaiotto–Creutzig conjecture on vertex operator superalgebras for S-duality

Let $\mathfrak g$ be a simple Lie algebra, let $P_n^+$ be the set of dominant weights $\lambda$ such that $n\lambda^2$ is integral, and let $\psi,\psi'$ be generic complex numbers satisfying

$$\frac{1}{\psi}+\frac{1}{\psi'}=n.$$

Define

$$A^n[\mathfrak g,\psi]:=\bigoplus_{\lambda\in P_n^+}\mathbb L_{\psi-h^\vee}(\lambda)\otimes\mathbb L_{\psi'-h^\vee}(\lambda).$$

Gaiotto–Creutzig conjecture. The object $A^n[\mathfrak g,\psi]$ can be given the structure of a simple vertex operator superalgebra.

This conjecture predicts vertex operator superalgebras associated with intersections of Dirichlet boundary conditions and their general $S$-duals. They are proposed as quantum geometric Langlands kernel vertex operator superalgebras; the source does not state a resolution.