Monotonicity conjecture for rational q-Catalan numbers

For coprime positive integers $a$ and $b$, define the rational $q$-Catalan number

$$\operatorname{Cat}(a,b)_q=\frac{1}{[a]_q}\genfrac{[}{]}{0pt}{}{a-1+b}{a-1}_q.$$

Here $[n]_q=(1-q^n)/(1-q)$ and $\mathbb{N}[q]$ denotes polynomials in $q$ with non-negative integer coefficients. Monotonicity conjecture. For any $a,b,c\geq 1$ with $\gcd(a,b)=\gcd(a,c)=1$ and $b<c$,

$$\operatorname{Cat}(a,c)_q-\operatorname{Cat}(a,b)_q\in\mathbb{N}[q].$$

This conjecture predicts coefficientwise monotonicity of rational $q$-Catalan numbers as the second parameter increases. The paper proves it for the infinite family of triples with $a\leq 20$, but the general case remains open.