Compactness of the group of invertible twisted representations

Let $\mathcal{A}$ be a completely rational conformal net, let $G\leq \operatorname{Aut}(\mathcal{A})$ be closed, and let $I_0$ be an interval. Write

$$\pi_{0}\bigl(G\text{--}\operatorname{Rep}^{I_0}(\mathcal{A})^{\times}\bigr)$$

for the group of equivalence classes of invertible $G$-twisted representations of $\mathcal{A}$ that are $G$-localized in $I_0$. Let $G_0\leq G$ be the image of the degree map $\partial$. Then there is a short exact sequence

$$1\to \pi_{0}\bigl(\operatorname{Rep}^{I_0}(\mathcal{A})^{\times}\bigr)\to \pi_{0}\bigl(G\text{--}\operatorname{Rep}^{I_0}(\mathcal{A})^{\times}\bigr)\to G_0\to 1.$$

Compactness claim. The group $\pi_{0}\bigl(G\text{--}\operatorname{Rep}^{I_0}(\mathcal{A})^{\times}\bigr)$ is compact.

The claim follows in the source from the displayed short exact sequence and complete rationality, but the supplied excerpt does not provide the details of the compactness argument or an independent resolution status.