The e-positivity conjecture for theta graphs

For any integers $a\ge b\ge c\ge 1$, let $\theta_{abc}$ be the theta graph formed from two distinct vertices joined by three pairwise disjoint paths of lengths $a$, $b$, and $c$. A graph is e-positive when its chromatic symmetric function has an expansion in the elementary symmetric-function basis with nonnegative coefficients. Theta-graph e-positivity conjecture. The theta graph $\theta_{abc}$ is $e$-positive.

This conjecture extends the classical e-positivity results for paths and cycles to theta graphs, which have three internally disjoint paths between the same pair of vertices. Its resolution is not given in the supplied text.