The localizability conjecture for simple objects in braided fusion categories

Let $\mathcal{C}$ be a braided fusion category and let $X$ be a simple object of $\mathcal{C}$. Denote by $\rho_X$ the braid-group representations on $\operatorname{End}(X^{\otimes n})$. Localizability conjecture. The representations $(\rho_X,\operatorname{End}(X^{\otimes n}))$ are localizable if and only if

$$\operatorname{FPdim}(X)^2\in\mathbb{N}.$$

This conjecture combines the property $\mathbf{F}$ conjecture with the finite-image conjecture for unitary finite-order Yang–Baxter solutions, relating local quantum realizations of braid representations to weak integrality of Frobenius–Perron dimensions. The source describes recent progress and further evidence, but does not state a resolution.