Commutant and fixed-point-free extension conjecture beyond collapsing levels

Let $n\geq 4$ and $r\geq 0$, set $\mathfrak{g}=\mathfrak{sl}(n+r+2|r)$, and let $\Pi$ be a rank-one Heisenberg vertex operator algebra. Define

$$A=\mathcal{W}_{-2}(\mathfrak{sl}(n+r+2|r),\theta),\qquad B=L_{-1}(\mathfrak{gl}(n+r|r))=L_{-1}(\mathfrak{sl}(n+r|r))\otimes\Pi,$$

and

$$C_{n,r}=\operatorname{Com}(B,A).$$

Let $h^\vee$ be the dual Coxeter number of $\mathfrak{sl}_{n-2}$, set $\ell=-h^\vee+\frac{n-1}{n}$, and let $f_{\mathrm{prin}}$ be a principal nilpotent element of $\mathfrak{sl}_{n-2}$. Beyond-collapsing-levels conjecture. The vertex operator algebras satisfy

$$C_{n,r}\cong\mathcal{W}_{\ell}(\mathfrak{sl}_{n-2},f_{\mathrm{prin}}),$$

and $A$ is a fixed-point-free simple-current extension in the category of ordinary modules of

$$L_{-1}(\mathfrak{sl}(n+r|r))\otimes\Pi\otimes\mathcal{W}_{\ell}(\mathfrak{sl}_{n-2},f_{\mathrm{prin}}).$$

This conjecture identifies a commutant with a principal $\mathcal{W}$-algebra and predicts a fixed-point-free simple-current extension structure beyond the collapsing levels. The supplied text does not state a resolution.