Conformal coset embedding conjecture for type A W-algebras

Let $\lambda=(\lambda_1\leq\dots\leq\lambda_n)\vdash N$, let $N_i=N-\sum_{j>i}\lambda_j$, and let $C^{k_i^\sharp}(\mathfrak{sl}_{N_i},f_{\widehat{\lambda}_i})$ be the affine coset associated with the hook-type W-algebra for $\widehat{\lambda}_i=(1,\dots,1,\lambda_i)$. Let $\pi$ be a rank-one Heisenberg vertex algebra. Conformal embedding conjecture. At generic levels $k$, for suitable levels $k_i^\sharp$ satisfying the level relations in the source, there is a conformal embedding

$$\bigotimes_{i=1}^n C^{k_i^\sharp}(\mathfrak{sl}_{N_i},f_{\widehat{\lambda}_i})\otimes\pi^{\otimes(n-1)}\hookrightarrow\mathcal{W}^k(\mathfrak{sl}_N,f_\lambda).$$

If $k=-N+p/q$ is admissible and $f=f_{r,q^s}$ is exceptional for $k$, with $0<r\leq q$, there is also a conformal embedding

$$\bigotimes_{\ell=0}^{s}\mathcal{W}_{-(p-N)+\frac{p-\ell q}{p-(\ell+1)q}}(\mathfrak{sl}_{p-N})\otimes\mathcal{W}_{-r+\frac{p-sq}{q}}(\mathfrak{sl}_{r})\otimes\pi^{\otimes s}\hookrightarrow\mathcal{W}_k(\mathfrak{sl}_N,f_{r,q^s}).$$

The conjecture refines the reduction-by-stages picture by decomposing type A W-algebras into affine cosets and Heisenberg factors, with a further exceptional-level form.