The eight-vertex bound for forbidden GaTEx subgraphs

Let $\mathfrak{F}_{\mathrm{GT}}$ be the set of forbidden subgraphs of GaTEx graphs, and let $F\in\mathfrak{F}_{\mathrm{GT}}$. The preceding discussion shows that every such forbidden subgraph has at least five vertices.

Eight-vertex bound. For all $F\in\mathfrak{F}_{\mathrm{GT}}$, we have

$$5\leq |V(F)|\leq 8.$$

The lower bound follows from the fact that forbidden subgraphs must be primitive and cannot be pseudo-cographs, while the upper bound is motivated by the possibility of two vertex-disjoint induced $P_4$s. The claim is presented as a conjectural bound and no resolution is supplied here.