Inverse Hamiltonian reduction embedding conjecture for type A W-algebras

Let $\lambda=(\lambda_1,\dots,\lambda_n)\vdash N$, let $a$ be the unique integer with $\lambda_{a-1}=1$ and $\lambda_a>1$, and define

$$\sigma(\lambda)=(\underbrace{1,\dots,1}_{a},\lambda_a-1,\lambda_{a+1},\dots,\lambda_n).$$

Set $m=\lambda_1+\dots+\lambda_a$, let $\Pi[0]$ be the rank-one Heisenberg-type vertex algebra, let $\Pi^{1/m}[0]$ be a vertex algebra extension of it, and let $\beta\gamma$ denote the beta-gamma system. Inverse reduction embedding conjecture. There exists an embedding

$$\mathcal{W}^k(\mathfrak{sl}_N,f_\lambda)\hookrightarrow\mathcal{W}^k(\mathfrak{sl}_N,f_{\sigma(\lambda)})\otimes\Pi^{1/m}[0]\otimes\beta\gamma^{\otimes(m-2)}.$$

The claim generalizes the known inverse Hamiltonian reduction embedding for hook-type W-algebras; the required extension $\Pi[0]\subset\Pi^{1/m}[0]$ is part of the assertion.