The simple vertex-superalgebra structure conjecture for A[sl_N,f,ψ]

Let $N=n+m$, let $f$ be any nilpotent element of $\mathfrak{sl}_N$, and let

$$A[\mathfrak{sl}_N,f,\psi]=\bigoplus_{\lambda\in P^+}V^k(\lambda)\otimes H_f(V^\ell(\lambda))\otimes V_{\sqrt N\mathbb Z+\frac{s(\lambda)}{\sqrt N}}.$$

Here $s(\lambda)$ is the lattice-label map defined in the source, and $k$ and $\ell$ are related to $\psi$ as there. Creutzig–Gaiotto's conjecture. For generic $k$, $A[\mathfrak{sl}_N,f,\psi]$ can be given the structure of a simple vertex superalgebra such that the top level of

$$V^k(\lambda)\otimes H_f(V^\ell(\lambda))\otimes V_{\sqrt N\mathbb Z+\frac{s(\lambda)}{\sqrt N}}$$

is odd for $\lambda=\omega_1,\omega_{N-1}$. This conjecture proposes a uniform simple vertex-superalgebra extension for arbitrary nilpotent reduction. The source gives no resolution status.