The Property F conjecture for Frobenius–Perron dimensions

Let $\mathcal{C}$ be a braided fusion category, and let $X,Y\in\mathcal{C}$ be simple objects. For each $n$, let

$$\rho_n^X(Y):\mathcal{B}_n\longrightarrow \operatorname{Aut}(\operatorname{Hom}(Y,X^{\otimes n}))$$

be the associated braid group representation. Property $F$ conjecture. The representation $\rho_n^X(Y)$ has finite image if and only if $\operatorname{FPdim}(X)^2\in\mathbb{Z}$. The source presents this as a structural conjecture for weakly integral categories and records evidence in several families.