Agoston et al.'s convex hull thrackle conjecture

Let $P$ be a set of $n$ points in general position in the plane, and let $C(\mathcal{S})$ be a family of distinct convex hulls of subsets of $P$ satisfying: no member contains another; every two members intersect; and the intersection of any three distinct members is contained in $P$. Such a family is a convex hull thrackle on $P$.

Agoston et al.'s conjecture. A convex hull thrackle on $n$ points has at most $n$ convex hulls.

The conjecture is false: the paper constructs, for every $n\geq 6$, a convex hull thrackle on $n$ points with $n+1$ convex hulls. It is nevertheless proved when the points are in convex position, and the paper establishes the general upper bound $2n$.