The power-of-\q conjecture for multiplicity-one weights

Let $\mathfrak{g}$ be a simple Lie algebra of rank $r$, and let $\lambda$ and $\mu$ be weights such that the multiplicity $m(\lambda,\mu)$ of $\mu$ in the irreducible representation with highest weight $\lambda$ is one. Let $m_q(\lambda,\mu)$ denote the associated $q$-multiplicity. Power-of-$q$ conjecture. If $m(\lambda,\mu)=1$, then

$$m_q(\lambda,\mu)=q^{f(r)},$$

where $f(r)$ is a function of the rank $r$ of the Lie algebra. The conjecture concerns the remaining multiplicity-one pairs not covered by the paper's computations; the paper provides evidence through explicit calculations and tables, but a proof is left for future work.