Generalized injectivity conjecture for GM-tree fraction labelings

Fix $\sigma\in \mathfrak S_3$, and let $k_1,k_2,k_3$ be nonnegative integers. Consider the generalized Markov tree $\mathrm{M}\mathbb T(k_1,k_2,k_3,\sigma)$. For any irreducible fractions $t,s\in[0,1]$, let $n_t$ and $n_s$ be $(k_1,k_2,k_3)$-GM numbers with fraction labelings $t$ and $s$. Generalized injectivity conjecture. If $n_t=n_s$, then $t=s$. The conjecture is motivated by examples in which equal generalized Markov values arise from distinct generalized Markov trees; the paper proposes injectivity of the fraction labeling within a fixed generalized tree as the appropriate generalization.