The hypertree F-positivity conjecture

Let $H=(V,E)$ be a hypertree, and let $X_H$ denote its chromatic symmetric function. The $F$-basis is the fundamental quasisymmetric-function basis. Hypertree F-positivity conjecture. For every hypertree $H$, $X_H$ is $F$-positive. This would follow from finding the required partition of the nonconstant colorings for every edge size; the result is established when all edge sizes are prime, and also for edge size $4$, but remains open in general.