Stanley's claw-free graph conjecture for chromatic symmetric functions

Let $G$ be a simple graph, and let $X_G({\bf x})$ denote its chromatic symmetric function. A graph is claw-free if it has no induced subgraph isomorphic to the claw graph $K_{1,3}$. A symmetric function is $s$-positive if it expands positively in the Schur function basis. Stanley's conjecture. If $G$ is claw-free, then $X_G({\bf x})$ is $s$-positive. Stanley verified the conjecture for co-bipartite graphs, and Gasharov proved it for claw-free incomparability graphs of posets; however, the examples in this paper give line graphs, hence claw-free graphs, whose chromatic symmetric functions are not $s$-positive, disproving the conjecture.