Payne–Wood's coloring conjecture for planar point sets

Let $P\subseteq\mathbb{R}^{2}$ be a set of $n$ points with at most $\sqrt{n}$ points collinear. A coloring of $P$ assigns a color to each point; a color class is in general position if it contains no three collinear points.

Payne–Wood's coloring conjecture. Every such set $P$ can be colored with $O(\sqrt{n})$ colors so that each color class is in general position.

This is presented as a strengthening of Payne and Wood's quadratic conjecture for the planar Ramsey number. The source does not state a resolution, so the conjecture remains open.