The abelian intertwining algebra conjecture for the sum of G-simple currents

Let $V$ be a vertex operator algebra and let $G$ be a group of automorphisms of finite order of $V$. A $G$-simple current is an irreducible weak $g$-twisted $V$-module whose tensor functor induces bijections between the relevant sets of irreducible twisted modules. Define

$$\bar{V}=\bigoplus M,$$

where the sum ranges over all $G$-simple currents of $V$. Abelian intertwining algebra conjecture. The object $\bar{V}$ is an abelian intertwining algebra in the precise sense of [DL]. This predicts that the collection of all $G$-simple currents carries the structure of an abelian intertwining algebra, extending the vertex operator algebra structure. The source provides no resolution, so the conjecture is recorded as open.