Reduction-by-stages conjecture for affine W-algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra, let $k\in\mathbf{C}$, and let $f_1,f_2$ be nilpotent elements satisfying the hypotheses of the source's Main Theorem on freeness. Let $\mathcal{W}^k(\mathfrak{g},f)$ be the affine W-algebra at level $k$, and let $\mathsf{L}\mathfrak{m}_0=\mathfrak{m}_0[t,t^{-1}]$. Affine W-algebra reduction-by-stages conjecture.

$$\mathcal{W}^k(\mathfrak{g},f_2)\cong\mathsf{H}^0_{f_0}(\mathsf{L}\mathfrak{m}_0,\mathcal{W}^k(\mathfrak{g},f_1)).$$

The conjecture is suggested for pairs of some hook-type nilpotent elements in $\mathfrak{sl}_n$ and predicts compatibility of Drinfeld–Sokolov reduction with reduction by stages. The source states that it has not yet been proved; recent constructions provide evidence and examples of inverse quantum Hamiltonian reduction.