Conjecture on Heisenberg cosets of regular reductions of W-algebras

Let $n,m$ be positive integers and let $p$ satisfy

$$m+1=p(n-1).$$

Let $\mathcal{W}_{n,m}$ be the family of W-algebras defined by quantum Hamiltonian reduction, let $H_{\mathrm{reg}}(\mathcal{W}_{n,m})$ be its regular quantum Hamiltonian reduction using the principal embedding of $\mathfrak{sl}_2$ in $\mathfrak{sl}_n$, and let $\mathcal H$ denote the Heisenberg vertex operator algebra. Narrow W-algebra coset conjecture. The Heisenberg coset is isomorphic to the narrow W-algebra of type $\mathfrak{sl}_n$ and parameter $p$:

$$\operatorname{Com}\left(\mathcal H,H_{\mathrm{reg}}(\mathcal{W}_{n,m})\right)\cong W^0(p)_{A_{n-1}}.$$

This proposes a uniform identification of these cosets with narrow W-algebras; no resolution of the general statement is given in the source.