Abelian sector conjecture for the figure-8 region

Let $X_8$ be the immersed figure-8 annulus in the sphere, let ${\mathcal C}_8$ be the set of superselection sectors detected by $X_8$, and let $d_\mu$ denote the quantum dimension of a sector $\mu$. Abelian sector conjecture. There exists a sector $\mu\in{\mathcal C}_8$ such that

$$d_\mu=1.$$

The conjecture is formulated for a reference state on $\mathbf{S}^2$; the disk version is equivalent by completing the disk to a sphere, whereas an annularly supported reference state need not have an Abelian sector. It is proved in the paper for Abelian anyon theories.