The two-step hook-reduction conjecture for type A W-algebras

Let $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_\ell)$ be a partition of $N$, and set

$$N_i=N-(\lambda_1+\lambda_2+\cdots+\lambda_i),\qquad \widehat{\lambda}_i=[\lambda_i,1^{N_i}].$$

Let $\mathrm{H}_{\mathbb{O}_{\widehat{\lambda}_i}}^0$ denote the BRST reduction associated with the hook-type partition $\widehat{\lambda}_i$, and let $\mathcal{V}^\mathsf{k}(\mathfrak{sl}_N)$ and $\mathcal{W}^\mathsf{k}(\mathfrak{sl}_N,\mathbb{O}_\lambda)$ be the corresponding affine and W-algebras. The two-step hook-reduction conjecture. There is an isomorphism of vertex algebras

$$\mathcal{W}^\mathsf{k}(\mathfrak{sl}_N,\mathbb{O}_\lambda)\simeq \mathrm{H}_{\mathbb{O}_{\widehat{\lambda}_\ell}}^0\cdots \mathrm{H}_{\mathbb{O}_{\widehat{\lambda}_1}}^0(\mathcal{V}^\mathsf{k}(\mathfrak{sl}_N)).$$

This conjecture asserts that type A W-algebras can be recovered by successive reductions associated with hook-type partitions; it was proposed and checked in small ranks, while the general statement remains open.