Equality conjecture for the Serre subcategories associated with a simple module

Let $L$ be a simple $\mathfrak{g}$-module. Let $\mathscr{A}(L)$ be the abelian category of finitely generated subquotients of modules $V\otimes_{\mathbb{C}}L$ with $V$ finite-dimensional. Let $\mathscr{B}(L)$ be the Serre subcategory of objects of Gelfand–Kirillov dimension strictly smaller than $\operatorname{GKdim}(L)$, and let $\mathscr{C}(L)$ be the Serre subcategory of objects annihilated by every indecomposable projective functor in the two-sided cell corresponding to $\operatorname{Ann}_{U(\mathfrak{g})}(L)$. Equality conjecture.

$$\mathscr{B}(L)=\mathscr{C}(L).$$

This equality would identify the dimension-theoretic negligible objects with those invisible to the annihilator cell, making the resulting Serre quotient a natural finite-length setting for projective-functor actions. The supplied context gives stability of these subcategories but no proof of equality.