Conjecture on conformal embeddings in the family of W-algebras

Let $n,m$ be positive integers with $n\geq 2$. Define $\mathcal{W}_{n,m}$ as the quantum Hamiltonian reduction of $V_k(\mathfrak{sl}_{n+m})$ at the level satisfying

$$k+n+m=\frac{n+m}{m+1}.$$

Here the reduction uses the principal embedding of $\mathfrak{sl}_2$ in the $\mathfrak{sl}_m$ subalgebra of $\mathfrak{sl}_m\oplus\mathfrak{gl}_n$. Conformal-embedding conjecture. The simple affine vertex operator algebra $L_{-\frac{nm+1}{m+1}}(\mathfrak{gl}_n)$ embeds conformally in $\mathcal{W}_{n,m}$. This is known for $m=1$, for $m=2$, and for $n=2$; the general case remains open.