Projective-generator pseudo-trace isomorphism conjecture for a vertex operator algebra

Let $\operatorname{Rep}(\mathcal{V})$ be the representation category of a vertex operator algebra $\mathcal{V}$, let $G$ be a projective generator of this category, let $E$ be its endomorphism algebra, and let $C(E)$ and $C_1(\mathcal{V})$ denote the spaces appearing in Proposition 4.1 of the source. Projective-generator pseudo-trace isomorphism conjecture. The map

$$C(E) \longrightarrow C_1(\mathcal{V})$$

from the stated proposition is an isomorphism of $\mathbb{C}$-vector spaces. This conjecture would identify all torus one-point pseudo-trace functions with central forms on the endomorphism algebra of a projective generator; it is used subsequently to transport modular transformations to $C(E)$ and remains open in the general non-semisimple setting.