The positive-coefficient numerator conjecture for Carlitz-Riordan polynomial interpolation

Let $C_n(x|q)$ be the polynomial defined in the paper, and let $[k]_q$ denote the $q$-integer. For each $n$, write $C_n(x|q)$ in lowest terms as a fraction with numerator $P_n(x|q)$ and denominator $[1]_q\cdots [n-1]_q$.

Positive-coefficient numerator conjecture. One can write

$$C_n(x|q)=\frac{P_n(x|q)}{[1]_q\cdots [n-1]_q},$$

where $P_n(x|q)$ has only positive coefficients.

This concerns the coefficient structure of the polynomial interpolation of Carlitz–Riordan's $q$-ballot numbers. The supplied text presents it among the paper's open problems, and gives no resolution.