Elekes's conjecture on collinear triples

Let $P\subset\mathbb R^2$ be a finite point set of size $n$. Elekes's collinear-triples conjecture. If $P$ determines $\Omega(n^2)$ collinear triples, then at least ten points of $P$ lie on a cubic. This strengthens the structural conclusion known for sufficiently many triple lines and remains open according to the survey.