Bergeron's positivity and unimodality conjecture for q-binomial differences

Let $a,b,c,d$ be positive integers such that $a$ is the smallest and $ad=bc$. The $q$-binomial coefficient is

$$\binom{m+n}{m}_q=\frac{(1-q)(1-q^2)\cdots(1-q^{m+n})}{(1-q)(1-q^2)\cdots(1-q^m)\cdot(1-q)(1-q^2)\cdots(1-q^n)}.$$

Thus $\binom{m+n}{m}_q$ is a symmetric polynomial in $q$. Bergeron's positivity and unimodality conjecture. The coefficients of the symmetric polynomial

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

are nonnegative and unimodal. This extends Bergeron's positivity problem, which asks whether these coefficients are always nonnegative. The conjecture is proved combinatorially in the cases $a\leq 3$ and $b,c\geq 4$; the general assertion remains open.