Barba's Ham-Sandwich line conjecture

Let $R$, $B$, and $G$ be sets of lines in $\mathbb{R}^3$ in general position, each containing an even number of lines. For two lines in general position, say that one lies above the other when the unique vertical line intersecting both visits the first and then the second when traversed from top to bottom. Barba's conjecture. There is a line $\ell$ in $\mathbb{R}^3$ that lies below exactly $|R|/2$ lines of $R$, $|B|/2$ lines of $B$, and $|G|/2$ lines of $G$; equivalently, a Ham-Sandwich line simultaneously bisects the three sets with respect to the above-below relation. This is the line-arrangement analogue of the Ham-Sandwich theorem and motivates the paper's generalization to mass assignments on subspaces. The supplied text gives no resolution status.