A counterexample to a conjecture of Grünbaum on piercing convex sets in the plane

A collection of sets FF has the (p,q)(p,q)-property if out of every pp elements of FF there are qq that have a point in common. A transversal of a collection of sets FF is a set AA that intersects every member of FF. Grünbaum conjectured that every family FF of closed, convex sets in the plane with the (4,3)(4,3)-property and at least two elements that are compact has a transversal of bounded cardinality. Here we construct a counterexample to his conjecture. On the positive side, we also show that if such a collection FF contains two disjoint compacta then there is a transversal of cardinality at most 13.