Rooted $(3+1)$-free $e$-positivity conjecture

Let $(P,\leq_P)$ be a poset, let $G(P)$ be its incomparability graph, and let $G(P)_*^r$ denote this graph rooted at $r\in P$. Let $X_0(G(P)_*^r)$ be the rooted chromatic symmetric function, and call it $e$-positive when each coefficient in its expansion as a polynomial in the root variable has an $e$-positive symmetric-function coefficient.

Rooted $(3+1)$-free $e$-positivity conjecture. For every $r\in P$, if $P-r$ is $(\mathbf{3}+\mathbf{1})$-free, then

$$X_0(G(P)_*^r)$$

is $e$-positive.

The authors report computer verification for posets with at most $8$ vertices, but the general assertion remains open.