Leray-complex sandwich problem

Determine whether, for a finite family F of convex sets in ℝᵐ with nerve K and a continuous map f:ℝᵐ→ℝᵈ, there exists a d-Leray complex L satisfying K⊆L⊆M, where M is the nerve of the family of images f(F).

Background

Given a family of convex sets and a continuous map, the nerve K of the original family is a subcomplex of the nerve M of the image family. The paper asks whether an intermediate d-Leray complex can always be found. An affirmative answer would reduce overlap Helly theorems to corresponding Helly theorems for d-Leray complexes.