Strict positivity of the coefficients 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 $m_{{\Greekmath 0115}}$ to the $q$-deformation of the exponential. Write

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

Positivity conjecture. For every partition ${\Greekmath 0115}$, all coefficients $a_k$ of $J_{{\Greekmath 0115}}(q)$ are strictly positive. This is one of three conjectures motivated by heuristic arguments and computations; the paper does not provide a proof or resolution.