The property F conjecture for braiding universality of anyons

An anyon $X$ has quantum dimension $d_x$.

Property F conjecture. An anyon $X$ is braiding universal if and only if

$$d_x^2\notin\mathbb{Z}.$$

This conjecture proposes an arithmetic criterion distinguishing braiding-universal anyons from those whose braid-group representations have finite image, the latter being called property F anyons. It has been verified in numerous cases, but the source does not establish it in full generality.