Dahlberg's $(e)$-positivity conjecture for labelled unit interval graphs

A labelled unit interval graph is a labelled graph satisfying the unit interval condition above. For a labelled graph $G$, write $y_G$ for its chromatic symmetric function and call $G$ $(e)$-positive when $y_G$ is $(e)$-positive.

Dahlberg's conjecture. All labelled unit interval graphs are $(e)$-positive.

This strengthens the Stanley–Stembridge conjecture because $(e)$-positivity implies ordinary $e$-positivity. Dahlberg verified the claim for all labelled unit interval graphs on at most $7$ vertices; the general case remains unresolved.