Additive coloring conjecture for 1-intersecting touching strings

A 1-intersecting set of strings is a set of strings in which any two strings intersect in at most one point, and it is $k$-touching if every point of the plane belongs to at most $k$ strings. Additive coloring conjecture. There is a constant $c>0$ such that every $k$-touching set of 1-intersecting strings can be colored with $k+c$ colors. No resolution is supplied in the source excerpt.