Quadratic-size conjecture for k-blocked point sets

A finite point set in the plane is $k$-blocked if its points are assigned one of $k$ colours such that two distinct points have the same colour exactly when some other point of the set blocks them. Quadratic-size conjecture. Every $k$-blocked point set has $O(k^2)$ points. The paper describes this as a strong conjecture; the largest known constructions are substantially smaller than quadratic, and no proof of the asserted upper bound is given.