Kawasetsu's coset coincidence conjecture

For integers $n\geq4$ and $k\geq0$, let $\mathcal C_k(n)$ be the simple quotient of the coset of $L_{k+1}(\mathfrak{gl}_{n-2})$ in $\mathcal W_k(\mathfrak{sl}_n,f_{\mathrm{min}})$, and let $\mathcal W_{k'}(\mathfrak{sl}_{2k+n},f_{\mathrm{prin}})$ be the principal type $A$ $\mathcal W$-algebra at level $k'$. Kawasetsu's conjecture. For all such $n$ and $k$,

$$\mathcal C_k(n)\cong\mathcal W_{k'}(\mathfrak{sl}_{2k+n},f_{\mathrm{prin}}),$$

where

$$k'=-(2k+n)+\frac{k+n-1}{k+n}\quad\text{or}\quad k'=-(2k+n)+\frac{k+n}{k+n-1}.$$

This is motivated by equality of central charges and is equivalent in the paper to an explicit conjectural description of the ideal defining the corresponding two-parameter quotient.