Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the ** local loop group**
$$
L_IG := \{\gamma\in LG \ | \ \forall z\not\in I \ \gamma(z)=e\}
$$
which is a subgroup of $LG$. Let $k\ge 1$ be an integer.
The level $k$ central extension of $LG$ is denoted $\mathcal{L}G_k$.
It restricts to a central extension of the local loop group that we denote $\mathcal{L}_IG_k$.

A representation of $\mathcal{L}G_k$ on a Hilbert space is called * positive energy* if it admits a covariant action of $S^1$ (i.e., the action should extend to $S^1\ltimes \mathcal{L}G_k$) whose infinitesimal generator has positive spectrum.
Here, the center of $\mathcal{L}G_k$ is required to act by scalar multiplication.

Definition 1:

Two level $k$ positive energy representation of the loop group are calledlocally equivalentif they become equivalent when restricted to $\mathcal{L}_IG_k$.

The follows is believed to be true:

Claim 2:Let $G$ be a cscsc group and let $V$ and $W$ be any two positive energy representations of $\mathcal{L}G_k$. Then $V$ and $W$ are locally equivalent.

~~I know a paper that proves the following:~~

Theorem 3:Let $G$ be asimply lacedcscsc group and let $V$ and $W$ be two positive energy representations of $\mathcal{L}G_k$. Then $V$ and $W$ are locally equivalent.

**Edit:** The argument in [GF] seems to contain a mistake (on lines -4 and -3 of
page 600)

~~The basic ingredients that are needed
(see page 599 of [Gabbiani & Fröhlich ~~*Operator algebras and conformal ﬁeld theory*] for the proof) are the following two facts about positive energy representations of simply laced loop groups:

• Every level 1 rep can be obtained from the vacuum rep by precomposing the action by an outer automorphism of $\mathcal{L}G_1$ that is the identity on $\mathcal{L}_IG_1$.

• Every level $k$ rep appears in the restriction of a level 1 rep under the map $\mathcal{L}G_k\to \mathcal{L}G_1$ induced by the $k$-fold cover of $S^1\to S^1$.

There are proofs in the literature, due to A. Wassermann (here p23) and V. Toledano-Laredo (here p82) respectively, for the cases $LSU(n)$ and $LSpin(2n)$, that are based on the theory of free fermions -- actually, Toledano only treats half of the representations of $LSpin(2n)$.

Is there a proof of Claim 2 in the literature?

How does one prove Claim 2?