Colouring multijoints

Let L_1, ..., L_d be pairwise disjoint collections of lines in a d-dimensional vector space over some field. If the collections are sufficiently generic we prove that there exists a d-colouring of the set of multijoints J such that for each j, for each line in L_j, the number of points on it of colour j is O(|J|^{1/d}).