Martínez–Roldán–Rubin transversal conjecture for pairwise intersecting convex sets

Let $\mathcal{F}$ be a finite family of convex sets in $\mathbb{R}^3$, with every two elements of $\mathcal{F}$ intersecting. A line transversal to a subfamily is a line intersecting every member of that subfamily. Martínez–Roldán–Rubin's transversal conjecture. There is a constant $\alpha>0$ such that $\mathcal{F}$ contains a subfamily $\mathcal{F}^*$ with

$$|\mathcal{F}^*|\geqslant \alpha|\mathcal{F}|$$

that has a line transversal. The conjecture asks for a positive-density line-transversal subfamily in every finite pairwise intersecting family of convex sets in three-dimensional space. The paper proves this conjecture for cylinders, with $\alpha=1/28$; the general case remains open.