Resolving Stanley's $e$-positivity of claw-contractible-free graphs

In Stanley's seminal 1995 paper on the chromatic symmetric function, he stated that there was no known graph that was not contractible to the claw and whose chromatic symmetric function was not $e$-positive, namely, not a positive linear combination of elementary symmetric functions. We resolve this by giving infinite families of graphs that are not contractible to the claw and whose chromatic symmetric functions are not $e$-positive. Moreover, one such family is additionally claw-free, thus establishing that the $e$-positivity of chromatic symmetric functions is in general not dependent on the existence of an induced claw or of a contraction to a claw.
Comment: Label corrected in Example 3.1, 28 pages, final version to appear in J. Eur. Math. Soc. (JEMS)