The equivariant reconstruction conjecture for VOA representation categories

Let $\mathcal{V}$ be a vertex operator algebra with symmetry group $G$, and let $G_0$ be the identity component of $G$. Let $\operatorname{Rep}(\mathcal{V})$ denote the representation category of $\mathcal{V}$. The equivariant reconstruction conjecture. The reconstruction map sends the symmetry group $G$ of $\mathcal{V}$ to $G/G_0$ for the representation category $\operatorname{Rep}(\mathcal{V})$. The source contrasts this proposed behavior with examples showing that the reconstruction map is not generally surjective or injective on symmetries; no proof or disproof is supplied.