Constant-additive coloring conjecture for pseudoline arrangements

Let $\mathcal{A}$ be an arrangement of pseudolines, and let $\operatorname{mx}(\mathcal{A})$ denote the maximum number of crossings on any pseudoline. A coloring of the crossings is proper along pseudolines if no color appears twice along any pseudoline. Constant-additive coloring conjecture. There is a constant $c$ so that the crossings of every pseudoline arrangement $\mathcal{A}$ can be colored using $\operatorname{mx}(\mathcal{A})+c$ colors so that no color appears twice along any pseudoline. This asks whether the chromatic requirement for arrangements of pseudolines differs from the maximum number of crossings on one pseudoline by at most an absolute additive constant.