The strongly e-positive graph characterization conjecture

Let $G$ be a finite graph. Call $G$ strongly $e$-positive if every induced subgraph $H$ of $G$ is $e$-positive, and call a graph $(\text{claw},\text{net})$-free if it contains neither the claw nor the net as an induced subgraph.

Strongly e-positive graph characterization conjecture. A graph $G$ is strongly $e$-positive if and only if it is $(\text{claw},\text{net})$-free.

The forward implication follows because neither the claw nor the net is $e$-positive. The reverse implication is open; all $(\text{claw},\text{net})$-free graphs with at most nine vertices were verified by computer to be $e$-positive, providing evidence for the conjecture.