Friedgut's iterated-shadow intersection conjecture

Let $n$ be even, let $\binom{[n]}{n/2}$ denote the middle layer of the Boolean cube, and let $\mathcal{A}\subset\binom{[n]}{n/2}$. Write $\mathcal{A}^c$ for its complement in this layer, let $\partial^{+r}(\mathcal{A})$ denote the $r$-fold upper shadow, and let $\mu$ be the uniform measure on the relevant layer. Friedgut's conjecture. For any $\zeta>0$ and $\epsilon>0$, there exists $\delta>0$ such that, if $\zeta\leq\mu(\mathcal{A})\leq1-\zeta$ and $r=\lceil\epsilon\sqrt{n}\rceil$, then

$$\mu\bigl(\partial^{+r}(\mathcal{A})\cap\partial^{+r}(\mathcal{A}^c)\bigr)\geq\delta.$$

The conjecture asserts that sets of uniformly nontrivial measure have iterated upper shadows with a quantitatively positive intersection after only order-$\sqrt n$ iterations. It was attributed in the source to Friedgut in personal communication; no resolution is given.