Piercing intersecting convex sets

Assume two finite families $\mathcal A$ and $\mathcal B$ of convex sets in $\mathbb{R}^3$ have the property that $A\cap B\ne \emptyset$ for every $A \in \mathcal A$ and $B\in \mathcal B$. Is there a constant $γ>0$ (independent of $\mathcal A$ and $\mathcal B$) such that there is a line intersecting $γ|\mathcal A|$ sets in $\mathcal A$ or $γ|\mathcal B|$ sets in $\mathcal B$? This is an intriguing Helly-type question from a paper by Martínez, Roldan and Rubin. We confirm this in the special case when all sets in $\mathcal A$ lie in parallel planes and all sets in $\mathcal B$ lie in parallel planes; in fact, all sets from one of the two families has a line transversal.