Equivalence of loop-group representation categories and conformal-net categories

Let $G$ be a simple simply connected Lie group and let $k\geq 0$ be a level. The category $\mathrm{Rep}^k(LG)$ is defined by approaches \textit{a.} or \textit{b.}, while approach \textit{c.} constructs another category; a balanced tensor category is a tensor category equipped with a braiding and a twist.

Equivalence conjecture. For every simple simply connected Lie group $G$ and every level $k\geq 0$, the categories $\mathrm{Rep}^k(LG)$ defined via \textit{a.} or \textit{b.}, and via \textit{c.}, are equivalent as balanced tensor categories.

Approaches \textit{a.} and \textit{b.} are known to produce equivalent modular tensor categories and to be additively equivalent to $\mathrm{Rep}^k(LG)$. Approach \textit{c.} is less developed: the equivalence is known for $G=SU(2)$, while for $SU(n)$ and for other Lie groups the corresponding braided or fusion-category equivalence was not known in the source.