Appendable $(e)$-positivity conjecture for labelled unit interval graphs

A labelled graph $G$ is $(e)$-positive if its chromatic symmetric function $y_G$ is $(e)$-positive. If $G$ and $H$ are labelled graphs, their concatenation $G+H$ is formed by identifying the final vertex of $G$ with the initial vertex of $H$. A labelled graph $H$ is appendable $(e)$-positive if $G+H$ is $(e)$-positive for every $(e)$-positive labelled graph $G$.

Appendable $(e)$-positivity conjecture. Every labelled unit interval graph is appendable $(e)$-positive.

This is explicitly described as an even more optimistic conjecture than the preceding $(e)$-positivity conjecture. The source gives no known general proof or disproof.