The property F conjecture for braided fusion categories

Let $\mcC$ be a braided fusion category. It has property F if the braid-group image $\rho_X(\mcB_n)$ is finite for every object $X\in\mcC$ and every $n$; equivalently, the associated braid-group representations on the centralizer algebras $\operatorname{End}_{\mcC}(X^{\otimes n})$ have finite image.

Property F conjecture. The category $\mcC$ has property F if, and only if,

$$\operatorname{FPdim}(\mcC)\in\mathbb Z,$$

that is, $\mcC$ is weakly integral.

This conjecture proposes a precise connection between finiteness of braid-group representations, relevant to topological quantum computation, and weak integrality of the Frobenius–Perron dimension. Its resolution status is not specified in the supplied text.