Ito's duality conjecture for regular \mathcal{W}-superalgebras

Assume $n\geqslant m$. Let $\mathcal{W}^{\check{\kappa}}(\mathfrak{gl}_{n},\mathbb{O}_{[n-m,1^m]})$ be the $\mathcal{W}$-algebra associated with the nilpotent orbit corresponding to the Jordan block $[n-m,1^m]$. For $\kappa,\check{\kappa}\notin\mathbb{Q}$ satisfying

$$(\kappa+n-m)(\check{\kappa}+n)=1,$$

set

$$\mathcal{F}_{n|m}=\begin{cases} V_{\mathbb{Z}^m}, & n>m,\\ \mathcal{A}^{m}\otimes V_{\mathbb{Z}^m}, & n=m,\end{cases}\qquad \check{\kappa}_{\circ}=\check{\kappa}+(n-m).$$

Ito's conjecture. There is an isomorphism of vertex superalgebras

$$\mathcal{W}^{\kappa}(\mathfrak{gl}_{n|m})\simeq \operatorname{Com}\left(V^{\check{\kappa}_{\circ}}(\mathfrak{gl}_{m}),\mathcal{W}^{\check{\kappa}}(\mathfrak{gl}_{n},\mathbb{O}_{[n-m,1^m]})\otimes\mathcal{F}_{n|m}\right),$$

where the $\widehat{\mathfrak{gl}}_{m}$-action is diagonal. This is a proposed generalization of the duality established in the $\mathfrak{gl}_{n|1}$ case; its validity beyond that setting is conjectural.