The induced claw-or-net conjecture for non-e-positive graphs

Let $G$ be a finite graph, and let $e$-positive mean that the chromatic symmetric function of $G$ has a nonnegative expansion in the elementary symmetric-function basis.

Induced claw-or-net conjecture. If $G$ is not $e$-positive, then $G$ contains an induced claw or an induced net.

This is the contrapositive of the reverse implication in the strongly $e$-positive graph characterization conjecture. The source presents that implication as challenging and reports computer verification of the corresponding $(\text{claw},\text{net})$-free graphs through nine vertices.