Sharpness conjecture for forbidden induced subgraphs and JEP decidability

Let $H$ be a graph, let $S$ be a finite set of graphs, and write $\operatorname{Forb}_\subseteq(S)$ for the graphs containing no member of $S$ as an induced subgraph. One-forbidden-graph sharpness conjecture. Problem JEP is decidable for all finite sets $S$ containing $H$ if and only if $H\subseteq P_4$. The paper proves the if direction; moreover, $H\subseteq P_4$ is equivalent to $\operatorname{Forb}_\subseteq(H)$ being well-quasi-ordered under induced subgraph containment. The converse direction remains conjectural.