The successive thick-hook reduction conjecture for classical W-algebras

Let $\mathfrak{g}$ be $\mathfrak{so}_N$ or $\mathfrak{sp}_N$, and let

$$\lambda=(\lambda_1^{m_1}\geq\lambda_2^{m_2}\geq\cdots\geq\lambda_\ell^{m_\ell})$$

be a partition parametrizing a nilpotent orbit, with $m_i\in\{1,2\}$ subject to the parity conditions for the relevant type. Define

$$N_i=N-(m_1\lambda_1+m_2\lambda_2+\cdots+m_i\lambda_i),\qquad \widehat{\lambda}_i^{m_i}=[\lambda_i^{m_i},1^{N_i}].$$

Let $\mathrm{H}_{\mathbb{O}_{\widehat{\lambda}_i^{m_i}}}^0$ be the corresponding BRST reduction. The successive thick-hook reduction conjecture. There is an isomorphism of vertex algebras

$$\mathcal{W}^\mathsf{k}(\mathfrak{g},\mathbb{O}_\lambda)\simeq \mathrm{H}_{\mathbb{O}_{\widehat{\lambda}_\ell^{m_\ell}}}^0\cdots \mathrm{H}_{\mathbb{O}_{\widehat{\lambda}_1^{m_1}}}^0(\mathcal{V}^\mathsf{k}(\mathfrak{g})).$$

The conjecture extends the successive hook-reduction proposal from type A to types B, C, and D; the general isomorphism is not established.