Bergeron's Gaussian-polynomial inequality conjecture

Let $a,b,c,d$ be positive integers satisfying

$$0<a\leq b<c\leq d,\qquad ad=bc.$$

The Gaussian polynomial $\binom{n}{k}_q$ is the $q$-analogue of the binomial coefficient, and $f(q)\geq g(q)$ means that $f(q)-g(q)$ has non-negative coefficients. Bergeron's conjecture. The difference

$$\binom{b+c}{b}_q-\binom{a+d}{a}_q$$

is symmetric and coefficientwise non-negative. The source presents this as the Gaussian-polynomial consequence of Vessenes' conjecture; it notes that symmetry is clear, while the non-negativity question was the subject of the paper's special cases and was not resolved in general there.