Feigin–Frenkel duality conjecture for regular W-superalgebras

Let $n,r\geq0$, let $\mathcal{W}^k(\mathfrak{sl}_{n+r|n},f_{n+r|n})$ be the universal regular W-superalgebra, let $\mathrm{SB}^{p|q}=\beta\gamma^{\otimes p}\otimes bc^{\otimes q}$, and let $V_{\mathbb{Z}^n}$ be the rank-$n$ lattice vertex algebra. W-superalgebra duality and rationality conjecture. For generic levels $k,\ell$ satisfying

$$(k+r)(\ell+n+r)=1,$$

there is an isomorphism

$$\mathcal{W}^k(\mathfrak{sl}_{n+r|n},f_{n+r|n})\simeq\begin{cases}\operatorname{Com}\left(V^{\ell+r}(\mathfrak{gl}_n),\mathcal{W}^{\ell}(\mathfrak{sl}_{n+r},f_{1^n,r})\otimes V_{\mathbb{Z}^n}\right),&r\geq1,\\operatorname{Com}\left(V^{\ell}(\mathfrak{gl}_n),V^{\ell}(\mathfrak{sl}_n)\otimes\beta\gamma^{\otimes n}\otimes V_{\mathbb{Z}^n}\right),&r=0.\end{cases}$$

Moreover, the regular W-superalgebras are rational and $C_2$-cofinite at levels $k=-r+r/p$ with $p\geq n+r$ and $r>n$. The conjecture connects resolved-conifold boundary-condition duality with rationality of regular W-superalgebras.