The beta-conjecture for Gaussian-polynomial inequalities

Let $a$ and $b$ be integers with $0<a<b$, and let $\beta\geq1$. The $\beta$-conjecture. One has

$$\binom{b+\beta a}{b}_q\geq\binom{a+\beta b}{a}_q,$$

where the inequality is coefficientwise for polynomials in $q$. This is the three-parameter specialization of the preceding Gaussian inequality. The source says that the paper proves the case $\beta=2$, while the conjecture for general $\beta\geq1$ is left open there.