Equivalence conjecture for connected-group loop-group representation categories

Let $G$ be a connected Lie group and let $k\in H^4(BG,\mathbb{Z})$ be a positive level. Consider the categories $\operatorname{Rep}^k(LG)$ defined via vertex operator algebras and via conformal nets. Equivalence conjecture. These categories are equivalent as balanced tensor categories. This extends the preceding conjecture from simple simply connected groups to connected groups; the vertex-operator-algebra and conformal-net constructions are available in this setting through simple-current extensions. The source gives no resolution in the stated generality.