Root-count conjecture for mutations in admissible Dynkin biagrams

Let $(\Gamma, \Delta)$ be an admissible Dynkin biagram with $r$ vertices. Let $h_{\Gamma}$ and $h_{\Delta}$ denote the Coxeter numbers associated to the respective components, and consider one period of the tropical $T$-system. A mutation is a $\Gamma$-mutation or a $\Delta$-mutation according to whether the corresponding maximum is attained by the $\Gamma$- or $\Delta$-sum. Root-count conjecture. The number of $\Gamma$-mutations in one period is the number of roots in the root systems associated to each $\Gamma$-component, totaling $h_{\Gamma}\cdot r$. Similarly, the number of $\Delta$-mutations in one period is the number of roots in the root systems associated to each $\Delta$-component, totaling $h_{\Delta}\cdot r$. The conjecture has been verified by Sage for all non-ADE families in the paper's classification for small $n$, as well as for all infinite ADE families for small $n$; these computations do not constitute a general proof.