The equivalence of local normality and positive energy for based loop groups

Let $\widetilde{\Omega G}$ be the centrally extended based loop group, and let $H$ be a representation of it at level $k$. Call $H$ locally normal when it belongs to the category of locally normal representations introduced in the paper, and call it positive energy when the representation extends to $\mathbb{R}\ltimes\widetilde{\Omega G}$ with energy-momentum generator $P$ having positive spectrum. Local-normality/positive-energy conjecture. A representation of the centrally extended based loop group is locally normal if and only if it has positive energy. This is stated as a strengthening of the preceding conjecture that locally normal representations have positive energy. The preceding implication was proved by Del Vecchio, Iovieno, and Tanimoto, while the converse and hence the stated equivalence are presented here as the stronger conjecture.