Uniqueness conjecture for generalized Markov numbers

For each pair $(q,p)in \mathbb Z_{>0}^2$ with $q\geq p$, let $m_{q,p}$ denote the generalized Markov number associated with $(q,p)$. Uniqueness conjecture for generalized Markov numbers. For any $(q,p),(q',p')\in \mathbb Z_{>0}^2$ with $q\geq p$ and $q'\geq p'$, if $(q,p)\neq(q',p')$, then

$$m_{q,p}\neq m_{q',p'}.$$

This conjecture asserts that distinct admissible pairs determine distinct generalized Markov numbers. The source proposes it after discussing monotonicity, but gives no resolution evidence.