The triple-line conjecture for point sets

Let $\CH$ be a finite set of points, and let $\harompt{\CH}$ denote the set of triple lines determined by $\CH$; write $|\CH|$ for the number of points in $\CH$. For a constant $c$, suppose that

Triple-line conjecture. If

$$|\harompt{\CH}| \geq c|\CH|^2,$$

then ten or more points of $\CH$ lie on a possibly degenerate cubic, provided that $|\CH|>n_0(c)$.

This conjecture asserts that a quadratic number of triple lines forces a substantial subset of the points onto a cubic curve, including degenerate cubics. The statement is presented as a natural expectation based on the examples discussed in the paper; its resolution is not specified in the supplied text.