Zanello's unimodality strengthening of Bergeron's Gaussian inequality

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

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

Consider the symmetric Gaussian-polynomial difference

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

Zanello's conjecture. Its coefficients are non-negative and unimodal. The source describes this as a strengthening of the preceding Gaussian inequality and notes that Zanello treated the special case $a\leq3$; the general assertion remains open in the supplied text.