Representation-category conjecture for superselection sectors of the quantum double model

Let $G$ be a finite group. Let $4\Delta4$ be the category of localised and transportable endomorphisms of the quantum double model, and let $4\operatorname{Rep}_f \mathcal{D}(G)4$ be the representation category of the quantum double $\mathcal{D}(G)$. Representation-category conjecture. There is an equivalence of categories

$$\Delta \simeq \operatorname{Rep}_f \mathcal{D}(G).$$

Such an equivalence would identify the superselection sectors of the quantum double model with finite-dimensional representations of the quantum double. The source presents this as a conjecture motivated by the relation between localised endomorphisms, amplimorphisms, fusion and braiding; it remains open there.