Type A coset conjecture for subregular W-algebras

Let $n\geq 4$ and $k\geq 0$ be integers satisfying

$$n^2-k^2-1\geq n.$$

Define

$${\mathcal C}_k(n)=\operatorname{Com}(L_{k+1}({\mathfrak {gl}}_{n-2}),{\mathcal W}_k({\mathfrak {sl}}_n,e_{-\theta})),$$

and set

$$\ell=-\frac{1+k^2+kn}{k+n}.$$

Type A coset conjecture. There is an isomorphism

$${\mathcal C}_k(n)\cong\operatorname{Com}({\mathcal H},{\mathcal W}_{\ell}({\mathfrak {sl}}_{k+1},f_{\operatorname{subreg}})).$$

For $k=0$ and $k=1$, the ${\mathcal W}$-algebra on the right is to be replaced by the rank-one $\beta\gamma$-system and the simple affine vertex algebra $L_{\ell}({\mathfrak {sl}}_2)$, respectively.

The proposed identification is motivated by matching central charges and the rarity of vertex algebras with the relevant central charge and one-dimensional associated variety. The general assertion remains open.