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

Let $\mathcal{K}$ be a finite family of $n$ convex sets in $\mathbb{R}^3$, and suppose that every two members $K_1,K_2\in\mathcal{K}$ satisfy $K_1\cap K_2\neq\emptyset$. Fractional line-transversal conjecture. There is an absolute constant $c>0$ such that a line crosses $cn$ members of $\mathcal{K}$. This is the one-colored variant highlighted by Bárány and Kalai; the source presents it as a subsequent open problem related to the 2-colored conjecture.