Reiner–Stanton conjecture on unimodality of symmetric q-binomial differences

Let $k$, $m$, and $b$ be integers with $k\ge 2$, $m\ge k$, and, for $k\ge 3$, $k-2\le b\le \frac{km-4k+4}{k-2}$, where $k(m-b)$ is even. Define

$$f(k,m,b)(q)=\binom{m}{k}_q-q^{\frac{k(m-b)}{2}+b-2k+2}\binom{b}{k-2}_q.$$

Here $\binom{m}{k}_q$ denotes the $q$-binomial coefficient. Reiner–Stanton conjecture. The polynomial $f(k,m,b)(q)$ is nonnegative and unimodal for all $m\gg_k0$ and all $b\le \frac{km-4k+4}{k-2}$ satisfying $kb\equiv km\pmod 2$, with the only exception of $b=\frac{km-4k+2}{k-2}$ when this is an integer.

The conjecture generalizes the original Reiner–Stanton prediction about symmetric differences of $q$-binomial coefficients. The paper proves the case $k=5$ and completely characterizes nonnegativity and unimodality for $k\le 5$; it also notes an isolated counterexample to the original conjecture when $k=3$. The cases $k\ge 6$ remain open in general.