Nonnegativity conjecture for the $q$-binomial quotient $G_{k,n}(q)$

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

$$F_{k,1}(q)=\sum_{n\ge 0}G_{k,n}(q)q^{n+1},\qquad G_{k,n}(q)=\frac{1}{1-q^{2n+1}}\frac{\begin{bmatrix}2k+2n-2\\2n\end{bmatrix}_{q}}{\begin{bmatrix}k+n-1\ \end{bmatrix}_{q^2}}.$$

Here $\begin{bmatrix}a\b\end{bmatrix}_q$ denotes the $q$-binomial coefficient, and $A(q)\succeq0$ means that all coefficients of $A(q)$ are nonnegative. Nonnegativity conjecture for $G_{k,n}(q)$. One has $G_{k,n}(q)\succeq0$. This is a refinement of the $m=1$ positivity conjecture and would imply it; the source presents it as an open conjecture supported by the displayed quotient and computational evidence.