Unimodality of the specialized monomial symmetric-function polynomials

Let ${\Greekmath 0115}$ be an integer partition, and let $J_{{\Greekmath 0115}}(q)$ be the polynomial associated with the specialization of the monomial symmetric function $m_{{\Greekmath 0115}}$. Write

$$J_{{\Greekmath 0115}}(q)=\sum_{k=0}^d a_kq^k.$$

The coefficient sequence is unimodal when there is an index $j$, $0\leq j\leq d$, such that $a_0\leq\cdots\leq a_j\geq\cdots\geq a_d$. Unimodality conjecture. For every partition ${\Greekmath 0115}$, the polynomial $J_{{\Greekmath 0115}}(q)$ is unimodal. In the paper this is stated as a consequence that would follow if the positivity and log-concavity conjectures were true, rather than as an independently supported result; it remains unresolved here.