Pierson's conjecture that the Kromatic symmetric function distinguishes all graphs

Let $G$ be a graph, and let $fbar{X}_G$ denote its Kromatic symmetric function. Two graphs are isomorphic when they differ only by a relabeling of their vertices. Pierson's conjecture. There do not exist nonisomorphic graphs $G$ and $H$ with $\overline{X}_G=\overline{X}_H$. Equivalently, the Kromatic symmetric function distinguishes all graphs up to isomorphism. The Kromatic symmetric function contains more information than the chromatic symmetric function, and is known to determine the number of induced copies of several graphs; whether it distinguishes every graph remains open.