The representation-category equivalence for loop group nets

Let $G$ be the compact, simply connected Lie group and $k$ the level appearing in the representation categories. Let $\mathrm{Rep}(\cA_{G,k})$ be the category of representations of the conformal net, let $\mathsf{Vec}_{\mathrm{f}}$ be the category of finite-dimensional vector spaces, and let $\mathsf{Hilb}$ be the category of Hilbert spaces and bounded linear maps. Write $\mathrm{Rep}^k_{\mathrm{f}}(\hat \g)$ for the category of finite direct sums of irreducible level-$k$ integrable positive-energy representations of the affine Lie algebra $\hat \g$. The representation-category equivalence conjecture. There is a natural equivalence of categories

$$\mathrm{Rep}(\cA_{G,k})\cong \mathrm{Rep}^k_{\mathrm{f}}(\hat \g)\otimes_{\mathsf{Vec}_{\mathrm{f}}}\mathsf{Hilb}.$$

This is intended to identify all representations of the conformal net with Hilbert-space completions of finite sums of irreducible affine Lie algebra representations, excluding representations that are not direct sums of irreducibles. Equivalently, via the comparison functor to loop-group representations, it asserts that every level-$k$ positive-energy representation is locally normal.