Linear-in-ε lower bound at √n iterations for half-measure families

Prove that for every even integer n and every family A of n/2-element subsets of [n] with μ(A) = 1/2, for all ε > 0 with r = ⌈ε√n⌉, the intersection of the r-th iterated upper shadows of A and its complement A^c has measure at least a constant times ε, i.e., establish μ(∂⁺ʳ(A) ∩ ∂⁺ʳ(A^c)) = Ω(ε).

Background

Beyond proving positive intersection for many iterations, the authors conjecture a sharper quantitative bound in the symmetric (half-measure) case. Specifically, they propose that the intersection grows linearly in ε when the number of iterations is r = ⌈ε√n⌉.

They note that their random restriction approach introduces a quadratic dependence on ε, suggesting that new techniques may be required to achieve the conjectured linear dependence.