Subregular coset coincidence conjecture

For $n\geq4$ and $m\geq3$, let $\mathcal C_k(n)$ be the simple quotient of the Heisenberg coset in the subregular $\mathcal W$-algebra of $\mathfrak{sl}_n$. Subregular coset coincidence conjecture. One has

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

where

$$k=-n+\frac{m+n}{n-1},\qquad k'=-m+\frac{m+1}{m+n}.$$

The source attributes this conjecture to the physics literature; it is equivalent to an explicit description of the ideal defining the subregular coset quotient.