Piercing intersecting convex sets

Assume two finite families $\mathcal{A}$ and $\mathcal{B}$ of convex sets in ${\mathbb{R}}^3$ have the property that $A\cap B\mathrm{\ne }\mathrm{\varnothing }$ for every $A\in \mathcal{A}$ and $B\in \mathcal{B}$. Is there a constant $\gamma >0$ (independent of $\mathcal{A}$ and $\mathcal{B}$) such that there is a line intersecting $\gamma \mathrm{\mid }\mathcal{A}\mathrm{\mid }$ sets in $\mathcal{A}$ or $\gamma \mathrm{\mid }\mathcal{B}\mathrm{\mid }$ 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.