Markov Uniqueness conjecture for Markov triples

Let $(a,b,c)$ and $(a',b',c')$ be positive integer solutions of the Markov equation

$$x^2+y^2+z^2=3xyz.$$

Markov Uniqueness conjecture. If $\max(a,b,c)=\max(a',b',c')$, then there exists a permutation $\sigma\in\mathfrak{S}_3$ such that

$$(a,b,c)=\sigma(a',b',c').$$

This century-old conjecture asserts uniqueness of a Markov triple up to permutation at each value of the maximum coordinate, and is still open.