Böröczky extremality conjecture for monochromatic lines

Let $P$ be a non-collinear set of $n$ two-colored points, and suppose that no line contains more than $(1/2-\delta)n$ points, where $\delta>0$. Böröczky extremality conjecture. For every $\delta>0$ there exists $\epsilon>0$ such that the number of monochromatic lines determined by $P$ is at least $\epsilon n^2$. The Böröczky example has only one monochromatic line when half the points lie on a line, and the conjecture asserts that staying a fixed proportion below this threshold forces quadratically many monochromatic lines.