Exceptional-level conformal embedding conjecture for type A W-algebras

Let $N\in\mathbb{Z}_{>0}$ and let $k=-N+p/n$ be admissible. Write $\pi$ for a rank-one Heisenberg vertex algebra, and set $L_\bullet(\mathfrak{sl}_i)=\mathbb{C}$ for $i=0,1$. Exceptional embedding conjecture. If $sn+r=N$ and $r\geq0$, then

$$\mathcal{W}_k(\mathfrak{sl}_N,f_{1^r,n^s})\hookleftarrow\bigotimes_{\ell=0}^{s}\mathcal{W}_{p-\ell n,p-(\ell+1)n}(\mathfrak{sl}_{p-N})\otimes L_{-r+(p-sn)/n}(\mathfrak{sl}_r)\otimes\pi^{\otimes(s-\delta_{r,0})}.$$

If $sn+r=N$ and $n>r\geq0$, then

$$\mathcal{W}_k(\mathfrak{sl}_N,f_{r,n^s})\hookleftarrow\bigotimes_{\ell=0}^{s}\mathcal{W}_{p-\ell n,p-(\ell+1)n}(\mathfrak{sl}_{p-N})\otimes\mathcal{W}_{-r+(p-sn)/n}(\mathfrak{sl}_r)\otimes\pi^{\otimes(s-\delta_{r,0})}.$$

These embeddings are proposed as a consequence of reduction by stages and describe exceptional W-algebras through smaller W-algebras, affine factors, and Heisenberg factors.