Twisted intertwining operators for finite automorphism groups

Let $V$) be a simple vertex operator algebra satisfying the following conditions: $V_{(0)}=\mathbb{C}\mathbf{1}$, $V_{(n)}=0$ for $n<0$, and the contragredient $V'$ is equivalent to $V$ as a $V$-module; every grading-restricted generalized $V$-module is completely reducible; and $V$ is $C_{2}$-cofinite, meaning that

$$\dim V/C_{2}(V)<\infty,$$

where $C_{2}(V)$ is spanned by the elements $\operatorname{Res}_{x}x^{-2}Y(u,x)v$ for $u,v\in V$, and $Y$ is the vertex operator map. Let $G$ be a finite group of automorphisms of $V$. The twisted intertwining-operator conjecture. The twisted intertwining operators among the $g$-twisted $V$-modules for all $g\in G$ satisfy associativity, commutativity, and modular invariance. If true, this would provide the genus-zero and genus-one parts of the chiral orbifold conformal field theory associated with $(V,G)$ and would give the corresponding category of twisted modules a natural $G$-crossed braided tensor-category structure with additional properties. The conjecture remains unresolved in the supplied source.