The surjective homomorphism conjecture for $W^{(2)}_n$ and $\mathcal B_{n+1}$

For general $n$, let $W^{(2)}_n$ be the Feigin–Semikhatov algebra at level $k=-n^2/(n+1)$, let $\mathcal B_{n+1}$ be the subalgebra constructed in the $(1,n+1)$-theory, and let $\omega$ denote the homomorphism defined in the construction. Surjective homomorphism conjecture. There exists a surjective vertex algebra homomorphism extending $\omega$,

$$W^{(2)}_n\longrightarrow\mathcal B_{n+1},$$

at level $k=-n^2/(n+1)$. The homomorphism is proved in the source for the cases corresponding to $p\leq 5$, while its existence for general $n$ remains open.