Frobenius's uniqueness conjecture for Markov triples

A Markov number is a positive integer occurring in a Markov triple, that is, a triple of positive integers satisfying the Markov equation. A Markov triple is ordered here as $(a,b,c)$ with $a\leq b\leq c$. Frobenius's uniqueness conjecture. For any Markov number $c$, there exists a unique Markov triple $(a,b,c)$ such that $a\leq b\leq c$. The conjecture is presented as the original uniqueness conjecture underlying the paper's generalized formulation; the supplied text does not give a resolution status for this original statement.