Generalized monotonicity conjecture for q-Ramanujan sums

For positive integers $a,b$ and $k$, define

$$\operatorname{Cat}(a,b;k)_q=\frac{1}{[a+b]_q}\sum_{d\mid\gcd(a,b)}c_d(k)_q\genfrac{[}{]}{0pt}{}{a/d+b/d}{a/d}_{q^d},$$

where $c_d(k)_q=\sum_{s\mid\gcd(d,k)}\mu(d/s)[s]_q$. Here $[n]_q$ is the $q$-integer, $\mu$ is the Möbius function, and $\mathbb{N}[q]$ denotes polynomials with non-negative integer coefficients. Generalized monotonicity conjecture. Fix integers $g,k\in\mathbb{N}$. For any $a,b,c\geq1$ satisfying $\gcd(a,b)=\gcd(a,c)=g$,

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

This conjecture generalizes the coprime monotonicity conjecture and the non-negativity conjecture of Reiner, Stanton, and White. The source reports computational tests for $a\leq20$ and $b\leq c\leq80$; the general assertion remains open.