Aigner's ordering conjectures for Markov numbers

For coprime positive integers $p,q$ with $1\leq p<q$, let $m_{p/q}$ denote the Markov number indexed by the rational number $p/q\in[0,1]$.

Aigner's conjectures. The following ordering properties should hold:

1. Fixed Numerator: If $q<q'$ and $\gcd(p,q')=1$, then

$$m_{p/q}<m_{p/q'}.$$

2. Fixed Denominator: If $p<p'<q$ and $\gcd(p',q)=1$, then

$$m_{p/q}<m_{p'/q}.$$

3. Fixed Sum: If $0<i<p$ and $\gcd(p-i,q+i)=1$, then

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

These conjectures concern the total ordering of Markov numbers induced by their rational indexing. The paper states that the corresponding conjectures for $k$-Markov numbers are established, while the ordinary Markov-number formulation is attributed to Aigner and is described as weaker than Frobenius' unicity conjecture.