The fractional line-piercing conjecture for intersecting convex sets in three dimensions

Let $\mathcal{F}$ be a finite family of compact, pairwise intersecting convex sets in $\mathbb{R}^3$.

Fractional line-piercing conjecture. There exists a constant $c$ such that some line intersects at least $c|\mathcal{F}|$ members of $\mathcal{F}$.

This is proposed as a motivating higher-dimensional problem for extending the KKM method. No resolution is given in the supplied context.