Partial-fraction positivity conjecture for $G_{k,n}(q)$

For $k,n\in\mathbb{N}_0$, let $G_{k,n}(q)$ be defined by

$$F_{k,1}(q)=\sum_{r\ge 0}G_{k,r}(q)q^{r+1}.$$

The polynomials $z_{k,j}(q)$ are to be chosen for $0\le j\le n$. Partial-fraction positivity conjecture for $G_{k,n}(q)$. One has

$$G_{k,n}(q)=\sum_{j=0}^{n}\frac{z_{k,j}(q)}{1-q^{2j+1}},$$

where every $z_{k,j}(q)$ has nonnegative coefficients. The source presents this as a further conjecture suggested by the forms in the proof of its first theorem and by computational evidence; it would suffice to establish positivity of $F_{k,1}(q)$.