Successive quantum Hamiltonian reduction conjecture for type A W-algebras

Let $\lambda=(\lambda_1\leq\lambda_2\leq\dots\leq\lambda_n)\vdash N$, and set $\widehat{\lambda}_i=(1,\dots,1,\lambda_i)\vdash N_i$ with $N_i=N-\sum_{j>i}\lambda_j$. Write $V^k(\mathfrak{sl}_N)$ for the universal affine vertex algebra and $H_{f_{\widehat{\lambda}_i}}$ for quantum Hamiltonian reduction by the hook-type nilpotent indexed by $\widehat{\lambda}_i$. Successive reduction conjecture. For $\lambda\vdash N$, there is an isomorphism of vertex algebras

$$\mathcal{W}^k(\mathfrak{sl}_N,f_{\lambda})\simeq H_{f_{\widehat{\lambda}_{1}}}H_{f_{\widehat{\lambda}_{2}}}\dots H_{f_{\widehat{\lambda}_{n}}}\left(V^k(\mathfrak{sl}_N)\right).$$

Moreover, the functors $H_{f_\lambda}$ and $H_{f_{\widehat{\lambda}_{1}}}H_{f_{\widehat{\lambda}_{2}}}\dots H_{f_{\widehat{\lambda}_{n}}}$ from the Kazhdan--Lusztig category of $V^k(\mathfrak{g})$-modules to $\mathcal{W}^k(\mathfrak{sl}_N,f_\lambda)$-modules are naturally isomorphic. This conjecture proposes that arbitrary type A W-algebras can be obtained by iterating hook-type reductions, including compatibility of the resulting reduction functors.