Uniqueness conjecture for Markov fractions

Let $q$ be a Markov number, and call a rational number a Markov fraction when it is one of the Markov fractions defined in the paper; its denominator is the denominator in lowest terms. The interval under consideration is $[0,\frac{1}{2}]$. Uniqueness conjecture for Markov fractions. For each Markov number $q$, the interval $[0,\frac{1}{2}]$ contains at most one Markov fraction with denominator $q$. This is equivalent to the uniqueness conjecture for Markov numbers, which asserts that every Markov number is the maximum of exactly one ordered Markov triple; the proposition is stated as yet to be proved or refuted.