Generalized k-Markov uniqueness conjecture for all labels

Let $\mathcal{R}$ be the set of labels used for generalized $k$-Markov numbers, and let $m^{(k)}_{(p,q)}$ denote the generalized $k$-Markov number attached to $(p,q)\in\mathcal{R}$. Generalized $k$-Markov uniqueness conjecture. For all $(p,q),(p',q')\in\mathcal{R}$ and all $k\geq 0$,

$$m^{(k)}_{(p,q)}=m^{(k)}_{(p',q')}\quad\Longrightarrow\quad(p,q)=(p',q').$$

This is presented as a strengthening of Gyoda and Maruyama's conjecture, requiring distinct labels to produce distinct generalized $k$-Markov numbers for every nonnegative $k$. The source provides supporting computations but no resolution.