Conjecture that line graphs of $e$-positive spiders are $e$-positive

Let $S$ be a spider, and let $S_L$ denote its line graph, whose vertices correspond to the edges of $S$ and whose adjacency records incidence of edges in $S$. A graph is $e$-positive when its chromatic symmetric function is a nonnegative linear combination of elementary symmetric functions.

Line-graph conjecture. If a spider $S$ is $e$-positive, then its line graph $S_L$ is $e$-positive.

The source notes that this holds for all $e$-positive spiders known to the authors, but gives no proof in general. The conjecture is also motivated by the relationship between connected partitions of a spider and of its line graph.