Monotonicity conjecture for consecutive $q$-binomial quotients

For $k,n\in\mathbb{N}$ with $k\ge n+1$, let $G_{k,n}(q)$ be defined by

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

Write $A(q)\succeq0$ when all coefficients of $A(q)$ are nonnegative. Monotonicity conjecture for $G_{k,n}(q)$. For $n\in\mathbb{N}$ and $k\ge n+1$,

$$G_{k,n}(q)-G_{k+1,n-1}(q)\succeq0.$$

The source says that the symmetry of $q$-binomial coefficients and computational data suggest this refinement; it would imply nonnegativity of all the $G_{k,n}(q)$ and hence the Andrews–Bachraoui positivity conjecture.