Martínez–Roldán-Pensado–Rubin's 2-colored line-transversal conjecture

Let $\mathcal{K}=\mathcal{K}_1\uplus\mathcal{K}_2$ be a finite 2-colored family of convex sets in $\mathbb{R}^3$. Suppose that every $K_1\in\mathcal{K}_1$ and $K_2\in\mathcal{K}_2$ satisfy $K_1\cap K_2\neq\emptyset$. Martínez–Roldán-Pensado–Rubin's conjecture. There is an absolute constant $c>0$ such that, for some $i\in\{1,2\}$, at least $c|\mathcal{K}_i|$ members of $\mathcal{K}_i$ can be crossed by a single line. The conjecture is a strengthening of line-transversal results for pairwise intersecting convex sets in three dimensions, and the paper identifies it as an open problem.