Non-negativity conjecture for q-Catalan germs

For each integer $a\geq3$, let $\operatorname{Cat}((a;c))_q$ denote the $q$-Catalan germ associated with an integer $c$, as defined in the paper. q-Catalan germ positivity conjecture. For every integer $c$ with $1\leq c\leq(a-1)^2$ and $\gcd(a,c)=1$,

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

The germs refine rational $q$-Catalan numbers into more basic polynomials. Computations for $3\leq a\leq5$ show non-negative coefficients, while the general positivity assertion remains open.