Coefficient-denominator conjecture for a second basic hypergeometric ratio

Let ${}_2\phi_1$ denote the basic hypergeometric series, and define $\gamma'_n(a,b,c)$ by

$$\sum_{n\ge0}\gamma'_n(a,b,c)z^n=\frac{{}_2\phi_1(aq,b;cq;q,z)}{{}_2\phi_1(a,b;c;q,z)}.$$

Second coefficient-denominator conjecture. For every $n$, there is a polynomial $P'_n(a,b,c)$ in $a,b,c,q$ with integer coefficients such that

$$\frac{\gamma'_n(a,b,c)}{1-c}=\frac{P'_n(a,b,c)}{\prod_{k=0}^{n}(1-cq^k)^{\left\lfloor\frac{n+1}{k+1}\right\rfloor}}.$$

This is a parallel denominator-structure conjecture for another ratio of basic hypergeometric series; the supplied text gives no evidence that it has been resolved.