The dimension criterion for generalized and quasi-localizability

Let $\mcC$ be a braided fusion category and let $X\in\mcC$ be a simple object. Write $\dim(X)$ for the categorical dimension of $X$; an object is generalized or quasi-localizable when it admits the corresponding localization described in the paper.

Dimension criterion for localizability. Any simple $X\in\mcC$ is generalized or quasi-localizable if, and only if,

$$\dim(X)^2\in\mathbb Z.$$

Under additional assumptions, the forward implication is known from results relating braid-group images to endomorphism algebras, but the stated equivalence is presented as a more general belief and remains open.