The e-positivity conjecture for hat-chains

A hat-chain is the graph constructed by joining the indicated complete graphs and paths, as in the hat-chain construction preceding the conjecture; an expression is e-positive when its chromatic symmetric function has a nonnegative expansion in the elementary symmetric-function basis.

Hat-chain e-positivity conjecture. Every hat-chain is $e$-positive.

Hat-chains form a family of conjoined graphs, and the conjecture asserts a uniform positivity property for their chromatic symmetric functions. The supplied text does not indicate whether this assertion has been proved or disproved.