The graded Drinfeld-center conjecture for holonomy invariants

Let $\mathcal{C}$ be the $\operatorname{SL}_2(\mathbb{C})^*$-graded category described in the paper, let $\mathcal{D}$ be the doubled category constructed from $\mathcal{C}$, and let $\mathcal{Z}_{\operatorname{SL}_2(\mathbb{C})^*}(\mathcal{C})$ denote the corresponding graded Drinfeld center. Graded Drinfeld-center conjecture. There is an equivalence

$$\mathcal{Z}_{\operatorname{SL}_2(\mathbb{C})^*}(\mathcal{C})\cong\mathcal{D},$$

so that the surgery invariant from $\mathcal{D}$ can be interpreted as the state-sum invariant from $\mathcal{C}$. This is motivated by the analogous equivalence in the non-graded case and by the graded state-sum/surgery correspondence, but the paper notes difficulties arising from the non-semisimplicity of $\mathcal{C}$ and the grading action.