The cubic-curve conjecture for point sets with a piercing set

Let $P$ be a set of $n$ points in general position in the plane, meaning that no three points of $P$ are collinear, and let $R$ be a set of $n$ points disjoint from $P$. Cubic-curve conjecture. If every line determined by $P$ passes through a point of $R$, then $P\cup R$ is contained in a cubic curve. This is the paper's main conjecture; its resolution is not indicated in the supplied text.