On Bergeron's positivity problem for \(q\)-binomial coefficients (Q1753091)

Summary: F. Bergeron [3] recently asked the intriguing question whether \(\binom{b+c}{b}_q -\binom{a+d}{d}_q\) has nonnegative coefficients as a polynomial in \(q\), whenever \(a,b,c,d\) are positive integers, \(a\) is the smallest, and \(ad=bc\). We conjecture that, in fact, this polynomial is also always unimodal, and combinatorially show our conjecture for \(a\leq 3\) and any \(b,c\geq 4\). The main ingredient will be a novel (and rather technical) application of Zeilberger's KOH theorem [15].