Compact orbifold uniqueness conjecture for unitary vertex operator algebras

Let $V$ be a simple unitary vertex operator algebra, let $(\cdot\mid\cdot)$ be its unitary scalar product, and let $G$ be a closed subgroup of $\operatorname{Aut}_{(\cdot\mid\cdot)}(V)$. Write $V^G$ for the fixed-point subalgebra and regard $V$ as a unitary $V^G$-module.

Compact orbifold uniqueness conjecture. The unitary $V^G$-module $V$ has a unique unitary VOA structure up to isomorphism.

This conjecture would imply that, when the partition function subalgebra is an orbifold fixed-point algebra, the original VOA is determined by its partition function. The paper proves an analogous uniqueness statement for simple strongly local VOAs, while the general unitary VOA claim remains open.