The branching-function conjecture for partition q-series of mutation loops

Let $\ell greater than or equal to 2$, let $\gamma=\gamma(X_r,\ell)$ be the mutation loop, and let $H_{\gamma}$ be its associated group. For $\sigma eed in H_{\gamma}$, let $\lambda eed in Q$ represent $F(\sigma)$, where $F$ is the stated isomorphism. Let $\mathcal{Z}_{\gamma}^{\sigma}(q)$ be the corresponding partition q-series, and let $b_{\lambda}^{0}(q)$ be the branching function. Branching-function conjecture.

$$q^{-\frac{c(\ell)-r}{24}}\mathcal{Z}_{\gamma}^{\sigma}(q)=b_{\lambda}^{0}(q).$$

This conjecture predicts that the normalized partition q-series coincides with the branching function for the vacuum highest weight. It is motivated by the conjectural branching-function formula cited in the source, but the supplied text gives no resolution.