Unicity conjecture for the generalized Markov values m(c(n,d))

Let $c(n,d)$ be the path associated with $(n,d)$ in the generalized Markov construction, and let $m(c(n,d))$ denote its associated value. Unicity conjecture. If

$$n\leq d,\qquad n'\leq d',\qquad m(c(n,d))=m(c(n',d')),$$

then

$$n=n'\quad\text{and}\quad d=d'.$$

Thus, within the region $n\leq d$, the values $m(c(n,d))$ should determine the pair $(n,d)$ uniquely. The source presents this as a stronger conjecture motivated by the observed repetitions and does not provide a resolution in the supplied text.