Colour-class-size conjecture for k-blocked point sets

Let $P$ be a $k$-blocked point set, with its points assigned one of $k$ colours so that two distinct points have the same colour exactly when some other point of $P$ blocks them. Colour-class-size conjecture. In every $k$-blocked point set there are at most $k$ points in each colour class. This is stated alongside the quadratic-size conjecture as a strong conjecture, and its resolution is not supplied in the paper.