The equivalence conjecture for the functor Phi between principal W-algebra module categories

Let $\chi:Z\to\Bbb C$ be a central character with core $c(\chi)$ of length $m$. Let $\cal P^{n-m}$ be the subcategory of $W^{n-m}$-modules which admit trivial generalized central character, and let

$$\Phi:\cal P^{n-m}\to W^\chi\text{-mod}$$

be the exact functor defined by $\Phi(M)=\operatorname{Res}_W(M\otimes V(\chi))$.

Equivalence conjecture. The functor $\Phi:\cal P^{n-m}\to W^\chi\text{-mod}$ defines an equivalence of categories.

The preceding lemma establishes that $\Phi$ is exact and sends simple objects to simple objects. The source does not provide evidence resolving whether this functor is an equivalence, so the claim remains open.