Wang's conjecture on e-positivity of theta graphs

For positive integers $a,b,c$, let $\theta_{abc}$ denote the theta graph formed by the union of three internally disjoint paths of lengths $a$, $b$, and $c$ with the same two distinct endpoints. A graph is $e$-positive if its chromatic symmetric function has an expansion in the basis of elementary symmetric functions with no negative coefficients.

Wang's conjecture. All theta graphs $\theta_{abc}$ are $e$-positive.

This conjecture generalizes the known $e$-positivity of cycle-chord graphs, which are the special case in which one of the three paths has length $1$. The source paper proves the clock-graph case $\theta_{ab2}$, but the full theta-graph conjecture remains open in the supplied context.