The weak-integrality conjecture for braid group representations

Let $\mathcal C$ be a braided fusion category and let $X$ be a simple object in $\mathcal C$. For each $n>0$, the braid group $\mathcal B_n$ acts on $\operatorname{End}(X^{\otimes n})$.

Weak-integrality conjecture. These braid group representations have finite image for all $n>0$ if and only if $\operatorname{FPdim}(X)^2\in\mathbb Z$.

This conjecture relates finiteness of braid group images to the Frobenius–Perron dimension of the simple object. The source notes that it is verified for one large class of weakly integral braided fusion categories, namely group-theoretical categories related to doubles of finite groups; the general statement remains open.