Equivalence conjecture for the vertex-operator-algebra tensor products

Let $V$ be a rational vertex operator algebra and let $[?]$ be its category of modules, equipped with the monoidal products $\underset{\mathbb{V}}{\otimes}$ and $\underset{{\text{\tiny HL}}}{\otimes}$. The first is constructed from conformal blocks, while the second is the Huang–Lepowsky tensor product defined using analytic methods and intertwining operators. The map

$$\mathcal{V}^S_{M,N} \overset{\cong}{\longrightarrow} \operatorname{Hom}_\mathbb{C}(\mathbb{V}_0(M,N,S'),\mathbb{C})$$

is natural in the modules $M,N,S$.

Tensor-product equivalence conjecture. The map above induces an equivalence of monoidal categories

$$(\mathcal{C}_V,\underset{\mathbb{V}}{\otimes})\simeq (\mathcal{C}_V,\underset{{\text{\tiny HL}}}{\otimes}).$$

The two tensor products are already naturally isomorphic and preserve the monoidal unit $V$; the remaining issue is compatibility of their associators. The expected equivalence is motivated by the fact that both associators arise from decompositions of a four-punctured sphere into three-punctured spheres, but the statement is presented as future work and is not established here.