The quantum Hamiltonian reduction conjecture for $W^{(2)}_n$

Let $n>1$, and let $\varphi_n:\mathfrak{sl}(2)\rightarrow\mathfrak{sl}(n)$ be an embedding such that, as $\mathfrak{sl}(2)$-modules, $\mathfrak{sl}(n)$ decomposes as

$$\lambda_{n-1}\oplus\lambda_{n-1}\oplus\bigoplus_{i=1}^{n-1}\lambda_{2i-1},$$

where $\lambda_m$ is the $m$-dimensional irreducible representation of $\mathfrak{sl}(2)$. Let $\mathcal W^{(2)}_n$ of level $k$ be the quantum Hamiltonian reduction of the affine vertex algebra of $\mathfrak{sl}(n)$ of level $k$ associated with $\varphi_n$, and let $W^{(2)}_n$ be the Feigin–Semikhatov algebra of level $k$. Quantum Hamiltonian reduction conjecture. For $n>1$, the two algebras $\mathcal W^{(2)}_n$ and $W^{(2)}_n$, both of level $k$, are isomorphic. This would identify the Feigin–Semikhatov algebras with quantum Hamiltonian reductions for the specified non-principal embeddings; the statement is presented as unproved in the source and remains open.