A coefficientwise Gaussian-polynomial product inequality

Let $a$ and $b$ be integers with $0\leq k\leq a<b$. The codicil conjecture. The following equivalent-looking coefficientwise inequalities hold:

$$\binom{a}{k}_q\binom{a+b}{b-k}_q\geq\binom{b}{k}_q\binom{a+b}{a-k}_q,$$

or

$$\binom{a}{k}_q\binom{b}{k}_q\binom{b+a}{b}_q\left[\frac{1}{\binom{a+k}{k}_q}-\frac{1}{\binom{b+k}{k}_q}\right]\geq0.$$

Here $\binom{n}{k}_q$ denotes the Gaussian polynomial and $\geq$ denotes coefficientwise comparison. The source presents this as a codicil arising from calculations related to the $\beta$-conjecture; no resolution is given in the supplied text.