Uniqueness conjecture for Markov-number continued fractions

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 $a<b$, write $b/a=[a_1,a_2,\ldots,a_n]$ as a continued fraction. 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 continued fraction expansion of $b/a$ contains only $1$s and $2$s.

The source states this as a stronger conjecture than the uniqueness conjecture involving Markov snake graphs; the source gives no resolution.