Cubic-curve extremal conjecture for blocking points

Let $P$ be a set of $n$ points in the plane in general position, and let $B$ be a set of blocking points for $P$, meaning that every line spanned by $P$ contains a point of $B$.

Cubic-curve extremal conjecture. If $|B|=n$, then all points in $P\cup B$ lie on a cubic curve.

This is an extremal variant of the blocking-points problem, motivated by the structure theorem for strong triangle blocking arrangements and by the extremal theorem of Green and Tao. The source does not provide evidence that the conjecture has been solved.