Canonical vertex-Ramsey characterization of the finite sunflower property

Let $\mathcal{K}$ be a relational Fraïssé class with strong amalgamation. The canonical vertex-Ramsey characterization.

$$\mathcal{K}\text{ has the canonical vertex-Ramsey property}\iff\mathcal{K}\text{ has the finite sunflower property}.$$

The source also conjectures that having the finite $2$-sunflower property is equivalent to having the finite sunflower property; it notes that the displayed equivalence would imply this. The question concerns whether the finite sunflower property admits a characterization analogous to that of the infinite sunflower property.