Parallel transportation conjecture for immersed annuli

Let ${\mathcal L}$ be an immersed annulus, let ${\mathcal C}_{\mathcal L}$ be the superselection sectors detected by it, and let $d_\mu$ be the quantum dimension of $\mu$. Transportation along ${\mathcal L}$ is parallel when $|{\mathcal I}({\mathcal L})|=|{\mathcal C}|$, where ${\mathcal I}({\mathcal L})$ is the set of sectors obtained by transportation and ${\mathcal C}$ is the set of anyon types. Parallel transportation conjecture. If transportation along ${\mathcal L}$ is parallel, then there exists $\mu\in{\mathcal C}_{\mathcal L}$ with

$$d_\mu=1.$$

The converse implication is established in the paper, so this conjecture is the remaining direction of the stated equivalence between parallel transportation and the existence of an Abelian sector.