Iterated quantum Hamiltonian reduction stable-equivalence conjecture

Let ${\mathfrak g}$ be a simple Lie algebra, let $({\mathfrak a}_1,\ldots,{\mathfrak a}_L)$ be ordered subalgebras whose direct sum is a subalgebra of ${\mathfrak g}$, and let $f_i$ be nilpotent elements of ${\mathfrak a}_i$. Set $f=f_1+\cdots+f_L$, and let

$$H\left(V^k({\mathfrak g})\otimes F({\mathfrak g},({\mathfrak a}_1,\dots,{ }_L),(f_1,\dots,f_L)),d_0\right)$$

be the total quantum Hamiltonian reduction. Two vertex algebras are stable equivalent when they become isomorphic after tensoring with suitable free-field algebras. Stable-equivalence conjecture. The vertex algebra

$$H\left(V^k({\mathfrak g})\otimes F({\mathfrak g},({\mathfrak a}_1,\dots,{ }_L),(f_1,\dots,f_L)),d_0\right)$$

and ${\mathcal W}^k({\mathfrak g},f)$ are stable equivalent. This proposes that the iterated reduction agrees with the ordinary reduction up to free fields; the source does not provide evidence resolving the conjecture.