Denominator conjecture for a basic hypergeometric coefficient ratio

Let ${}_{2}\phi_{1}(a,b;c;q,z)$ denote the basic hypergeometric series, and define coefficients $\gamma_n(a,b,c)$ by

$$\sum_{n\geq 0}\gamma_n(a,b,c)z^n=\frac{{}_{2}\phi_{1}(aq,bq;cq;q,z)}{{}_{2}\phi_{1}(a,b;c;q,z)}.$$

Denominator conjecture. For every $n\geq 0$,

$$\frac{\gamma_n(a,b,c)}{1-c}=\frac{P_n(a,b,c)}{\prod_{k=0}^n(1-cq^k)^{\left\lfloor\frac{n+1}{k+1}\right\rfloor}},$$

for some polynomial $P_n(a,b,c)$ in $a,b,c,q$ with integer coefficients.

The conjecture concerns the arithmetic form of coefficients arising from a ratio of contiguous basic hypergeometric series, in connection with $q$-continued fractions. Its resolution is not given in the supplied source context.