uniqueness conjecture for Markov numbers

Conjecture. Given a Markov number $z>1$, there are several other Markov numbers $x$ and $y$ such that $x^2+y^2+z^2=3xyz$, but there is only set of values of $x$ and $y$ satisfying the inequality $z>y\ge x$.

The conjecture is easy enough to check for small values. For example, for $z=5$, we could even test $x$ and $y$ we know not to be Markov numbers with minimum increase in computational overhead. Trying the triples in order: (1, 1, 5), (1, 2, 5), (1, 3, 5), (1, 4, 5), (2, 1, 5), … (4, 4, 5) against $15xy-(x^2+y^2+25)$ we obtain the sequence $-12$, 0, 10, 18, 0, 27, 52, 75, 10, 52, 92, 130, 18, 75, 130, 183. It doesn’t take significantly larger Markov numbers to show the need for a general proof of uniqueness. Many attempted proofs have been submitted, but Richard Guy dismisses them all as seemingly faulty.

A divide-and-conquer approach to the problem has yielded encouraging results, however. Baragar proved the uniqueness of prime Markov numbers $p$ as well as semiprimes $2p$. Schmutz then proved the uniqueness of Markov numbers of the forms $p^n$ and $2p^n$. Ying Zhang used these results to extend this to $4p^n$ and $8p^n$.