Center conjecture for critical-level regular \mathcal{W}-superalgebras

Assume $n\geqslant m$, with $\kappa_c$ the critical level, and let $\mathfrak{z}$ denote the commutative center. Let $\mathcal{A}^m_{\mathrm{deg}}$ be the degenerate $\beta\gamma$-system, with its natural $\mathrm{GL}_m$-action. Center conjecture. There are isomorphisms of commutative vertex algebras

$$\mathfrak{z}(V^{\kappa_c}(\mathfrak{gl}_{n|m}))\simeq\mathfrak{z}(\mathcal{W}^{\kappa_c}(\mathfrak{gl}_{n|m}))\simeq\mathbb{C}[J_\infty\mathbb{A}^{n-m}]\otimes(\mathcal{A}^m_{\mathrm{deg}})^{\mathrm{GL}_m}.$$

The proposed description follows from the critical-level regular $\mathcal{W}$-superalgebra conjecture and is motivated by the corresponding description of centers in the cited work.