A counterexample to the claw-free Schur-positivity conjecture

The claw-free Schur-positivity conjecture, recorded by Stanley (1998) and credited there to Gasharov, asserts that the chromatic symmetric function of every claw-free graph is Schur-positive. We give a counterexample on 12 vertices: the line graph $G$ of the graph obtained from a 4-cycle by attaching triangles at two opposite vertices and pendant edges at the other two satisfies $[s_{(3,3,3,3)}]X_G = -64$. The coefficient follows from a short computation by hand and is also reproduced by three exact implementations. An exhaustive computation over all 216,777 connected claw-free graphs on at most 11 vertices shows that every one is Schur-positive, so 12 vertices is the minimum order of any counterexample. A complete census of the 1,728,404 connected claw-free graphs on 12 vertices finds exactly two non-Schur-positive isomorphism classes; the other has graph6 code K?`CR@`bAbRB and coefficient $[s_{(3,3,3,3)}] = -40$.