Nichols-algebra conjecture for screening operators

Let $W$ be a vertex operator algebra containing Heisenberg fields $\alpha_i$, and let $S_i=\operatorname{Res}(e^{\alpha_i})$. Assume that $W$ decomposes into integer eigenvalues under the zero-modes of the $\alpha_i$. Define

$$M=\bigoplus_i \sigma_{\alpha_i}W,$$

where $\sigma_{\alpha_i}$ is the spectral flow generated by $\alpha_i$. Assume that each $\sigma_{\alpha_i}W$ has diagonal braiding and defines a finite-dimensional Nichols algebra. Let $V=\bigcap_i\operatorname{Ker}(S_i)$, let $A$ be the algebra object defined by $W$ in the category of $V$-modules, and let $\mathcal B(M)$ be the Nichols algebra of $M$ in $A\operatorname{-Mod}_{\mathrm{loc}}$. Nichols-algebra conjecture for screening operators. There is an equivalence of tensor categories

$$\mathcal B(M)\operatorname{-Mod}(A\operatorname{-Mod}_{\mathrm{loc}})\simeq A\operatorname{-Mod}$$

compatible with the central structure of $A\operatorname{-Mod}_{\mathrm{loc}}$ on both sides. The conjecture seeks a general categorical realization of the relation between screening operators and Nichols algebras; the source records supporting results for Nichols-algebra relations and classifications in the diagonal-braiding case, but does not state a resolution of the equivalence.