The primitive pure graph conjecture

Let a pure graph $G$ (or its parity class $[G]$) be primitive when, for every family $\mathcal{F}$ of pure classes, $[G]\in\overline{\mathcal{F}}$ implies $[G]\in\mathcal{F}$. A pure graph is prime when it has no non-trivial decomposition of the type considered in the paper. The primitive pure graph conjecture. All primitive pure graphs are prime. This is stated as a conjecture about the relationship between the closure-based notion of primitivity and graph primeness; the supplied text gives no resolution status.