Two-point Helly

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.