Overlap Alon–Kleitman (p,q) theorem

Establish whether, for all integers p≥q>d, there exists an integer t=t(p,q,d) such that every finite family F of convex sets in ℝⁿ with the (p,q) property has the following property: for every continuous map f:ℝⁿ→ℝᵈ, t fibers of f together intersect every member of F.

Background

The Alon–Kleitman (p,q) theorem asserts a bounded transversal phenomenon for families of convex sets satisfying the (p,q) property. The paper seeks an overlap analogue in which a bounded number of fibers of an arbitrary continuous map to ℝᵈ replaces a bounded number of points. The authors state that this is an overlap version they have been unable to achieve by their methods.