The crystal Temperley–Lieb center conjecture

Let $\mathbf{CrysTL}$ be the crystal Temperley–Lieb category, and let $\mathcal{Z}(\mathbf{CrysTL})$ denote its Drinfeld center. The category has a canonical braided tensor functor

$$G:\operatorname{Rep}(\mathbb{Z}/2\mathbb{Z})\to\mathcal{Z}(\mathbf{CrysTL}).$$

Crystal Temperley–Lieb center conjecture. The canonical functor $G$ is a monoidal equivalence. This conjecture asserts that the only objects in the Drinfeld center are those induced by the $\mathbb{Z}/2\mathbb{Z}$-grading; the source provides partial results but does not establish the equivalence.