Suppose that for a finite collection of planar convex sets $\mathcal F$ the following holds.

For any six members of $\mathcal F$ there are two points such that every set contains (at least) one of the points.

Does it follow that all members of $\mathcal F$ can be stabbed by two points?

I am sure that this is known, and probably even the more general problem of determining the optimal value instead of six for $k$ stabbing points in $\mathbb R^d$.

Related question with many links to related problems: A curious generalization of Helly's theorem.