Colorful four-point piercing conjecture for two families of circles

Let $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ be finite families of circles in $\mathbb{R}^{2}$. Suppose that $A \cap B \neq \emptyset$ for every $A \in \mathcal{F}_{1}$ and $B \in \mathcal{F}_{2}$.

Colorful four-point piercing conjecture. Either $\mathcal{F}_{1}$ or $\mathcal{F}_{2}$ can be pierced by $4$ points.

This is proposed as a colorful analogue of Danzer's four-point piercing theorem for pairwise intersecting families of circles. The source presents it as a future-work conjecture and gives no resolution.