Continuous second selection lemma for dense complexes

Determine whether, for every α∈(0,1] and integer d≥1, there exists s=s(α,d)>0 such that every simplicial complex K on n vertices with at least α times the total number of (d+1)-vertex subsets as d-dimensional faces has the s-overlap property in dimension d for the class of all continuous maps K→ℝᵈ.

Background

The second selection lemma gives overlap bounds for facewise linear maps on complexes whose d-skeleton is sufficiently dense. The paper asks whether an analogous bound holds for all continuous maps, which would in turn imply the proposed overlap fractional Helly theorem. The question is stated explicitly as a problem and is not resolved in the paper.