The fixed denominator conjecture for Markov numbers

Let $m_{p/q}$ denote the Markov number indexed by a reduced rational number $p/q$ with $0<p<q$. Fixed denominator conjecture. For positive integers $p,q,i$ such that $p+i<q$, $\gcd(q,p)=1$, and $\gcd(q,p+i)=1$, one has

$$m_{p/q}<m_{(p+i)/q}.$$

This conjecture orders Markov numbers with the same denominator by increasing numerator; it is presented alongside the fixed numerator conjecture as a main topic of the paper, and its status is not resolved in the supplied source.