Explicit $W^{(2)}_N$ subalgebras of extended $\W_{n,\ell}$-algebras

Let $\mathfrak{W}_{n,\ell}$ be an extended algebra constructed from $\widehat{\mathfrak{gl}}(1\mid 1)$, and let $W^{(2)}_N$ denote the Feigin–Semikhatov algebra at level $K$. Explicit subalgebra conjecture. The extended algebra $\mathfrak{W}_{n,\ell}$ has a subalgebra isomorphic to $W^{(2)}_N$ of level $K$ in each of the following cases:

$$\begin{array}{ll} \ell=1,\ n=0,1,2,\ldots, & N=2n+1,\quad K=-2(n-1)(2n+1)/(2n-1);\\ \ell=1,\ n=\tfrac12,\tfrac32,\tfrac52,\ldots, & N=2n+1,\quad K=-(2n^2-1)/n;\\ \ell=2,\ n=-\tfrac34,-\tfrac14,\tfrac14,\ldots, & N=4(n+1),\quad K=-2(n+1)(4n+1)/(2n+1);\\ n=-\tfrac12(\ell-1),\ \ell=1,2,3,\ldots, & N=\ell,\quad K=-(\ell^2-\ell-1)/\ell. \end{array}$$

These cases arise from matching the operator-product expansions of explicitly constructed generators with those of $W^{(2)}_N$; the source presents the result as a summary of its findings, but the parser supplies no resolution evidence for whether it is established as a theorem or remains conjectural.