Uniqueness conjecture for Markov-number sum-of-squares representations

Let $m>2$ be a Markov number, meaning an integer belonging to a triple of positive integers satisfying

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

For coprime positive integers $a<b$, the palindromification of the snake graph $\mathcal{G}(b/a)$ is the snake graph associated with the continued fraction obtained by mirroring the continued-fraction data for $b/a$. Uniqueness conjecture. For every Markov number $m>2$, there exist unique positive integers $a<b$ with $\gcd(a,b)=1$ such that

$$m=a^2+b^2,\qquad 2a\leq b<3a,$$

and the palindromification of $\mathcal{G}(b/a)$ is a Markov snake graph.

The preceding corollary establishes existence, while uniqueness is conjectured; the source gives no resolution.