Constant-line transversal conjecture for pairwise intersecting convex sets in three dimensions

Let $\mathcal{K}$ be a finite family of pairwise intersecting convex sets in $\mathbb{R}^3$. Constant-line transversal conjecture. The family $\mathcal{K}$ admits a transversal by $O(1)$ lines. The source lists this as a more tangible goal and an open problem under the hypotheses of its main theorem.