A variant of the Hadwiger-Debrunner (p, q)-problem in the plane

Let X be a convex curve in the plane (say, the unit circle), and let S be a family of planar convex bodies, such that every two of them meet at a point of X. Then S has a transversal N ⊂ R of size at most 1.75 · 10. Suppose instead that S only satisfies the following “(p, 2)-condition”: Among every p elements of S there are two that meet at a common point of X. Then S has a transversal of size O(p). For comparison, the best known bound for the Hadwiger–Debrunner (p, q)-problem in the plane, with q = 3, is O(p). Our result generalizes appropriately for R if X ⊂ R is, for example, the moment curve.