Piercing intersecting convex sets

Assume two finite families A and B of convex sets in R3 have the property that A ∩ B ≠ ∅ for every A ∈ A and B∈B. Is there a constant γ > 0 (independent of A and B) such that there is a line intersecting γ|A| sets in A or γ|B| sets in 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 A lie in parallel planes and all sets in B lie in parallel planes; in fact, one of the two families has a transversal by a single line.