Quantum-dimension conjecture for compact-group decompositions of vertex operator algebras

Let $V=(V,Y,1,\omega)$ be a rational, $C_2$-cofinite simple vertex operator algebra, and let $G$ be a subgroup of $\operatorname{Aut}(V)$ that is a finite-dimensional compact Lie group acting continuously on $V$. Write

$$V=\sum_{\chi\in\operatorname{Irr}(G)}W_\chi\otimes V_\chi.$$

Here $W_\chi$ denotes the irreducible $G$-module indexed by $\chi$, $V_\chi$ is the corresponding $V^G$-module, and $V^G$ is the fixed-point vertex operator algebra.

Quantum-dimension conjecture. For every $\chi\in\operatorname{Irr}(G)$,

$$\dim W_\chi=\operatorname{qdim}_{V^G}V_\chi.$$

Motivated by lattice vertex operator algebra examples, this conjecture proposes that the multiplicity-space dimensions in the compact-group decomposition are given by quantum dimensions over the fixed-point algebra. The source gives no resolution or restricted theorem covering the full stated generality.