Subregular W-algebra coset truncation conjecture

For $n\geq4$, let $\mathcal C^k(n)$ be the coset of the Heisenberg algebra in $\mathcal W^k(\mathfrak{sl}_n,f_{\mathrm{subreg}})$, and let $\mathcal W^{K_n}_{R_n}(c,\lambda)$ denote the localized two-parameter quotient defined by an ideal $K_n$ and a localization $R_n$. Subregular coset truncation conjecture. There exist $K_n$ and $R_n$ such that $\mathcal W^{K_n}_{R_n}(c,\lambda)$ has a singular vector of weight $2n+2$ of the form

$$W^{2n+2}-P(L,W^3,\dots,W^{2n}),$$

and its maximal proper graded quotient is isomorphic to $\mathcal C^k(n)$, with $c$ and $k$ related by the central-charge formula in the source. This would imply that $\mathcal C^k(n)$ is of type $\mathcal W(2,3,\dots,2n+1)$; this is known for $n=4$.