The localization conjecture for simple objects

Let $\mathcal{C}$ be a unitary braided fusion category, and let $X\in\mathcal{C}$ be simple. Localization conjecture. The object $X$ can be localized if and only if

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

The source says this has been verified for Jones representations of braid groups, but gives no general proof.