The braided tensor equivalence conjecture for Temperley–Lieb and Virasoro categories

Let $p$ be an integer and let $\mathfrak{q}=e^{i\pi/p}$. Let $\mathsf{C}_{\infty}$ be the direct limit of Temperley–Lieb module categories at this parameter, equipped with the braided tensor structure from the stated theorem. Let $\mathsf{Vir}_p$ be the braided tensor category of Virasoro algebra representations at central charge $c(p)$.

Temperley–Lieb–Virasoro braided equivalence conjecture. There is an equivalence of braided tensor categories

$$\mathsf{C}_{\infty}\xrightarrow{\quad\sim\quad}\mathsf{Vir}_p$$

that reduces to the previously established equivalence when both categories are regarded only as abelian $\mathbb{C}$-linear categories.

This conjecture upgrades the known abelian-category correspondence to a monoidal and braided one, including tensor products, associators, and braidings. The proposed tensor structure is motivated by vertex-operator-algebra theory and by physical computations of fusion rules, but the braided equivalence itself remains unproved in the supplied text.