The root-count conjecture for colored mutations of admissible Dynkin biagrams

Let $(\Gamma, \Delta)$ be an admissible Dynkin biagram with $r$ vertices. Its tropical $T$-system has red mutations, called $\Gamma$-mutations, and blue mutations, called $\Delta$-mutations, according to which summand realizes the maximum in the tropical mutation rule. Let $h_{\Gamma}$ and $h_{\Delta}$ denote the Coxeter numbers associated to the root systems of the $\Gamma$- and $\Delta$-components, respectively.

Colored-mutation root-count conjecture. The number of $\Gamma$-mutations in one period is the total number of roots in the root systems associated to the $\Gamma$-components, $h_{\Gamma}\cdot r$. Similarly, the number of $\Delta$-mutations in one period is the total number of roots in the root systems associated to the $\Delta$-components, $h_{\Delta}\cdot r$.

This predicts that the two colors occurring during one period are counted by the corresponding root systems. The statement is presented as a pattern observed in previous work; no resolution is supplied in the source.