The extremal half-sized iterated-shadow intersection conjecture

Let $n$ be even and let $\mathcal{A}\subset\binom{[n]}{n/2}$ satisfy $\mu(\mathcal{A})=1/2$, where $\mu$ is the uniform measure on the middle layer, $\mathcal{A}^c$ is the complement in that layer, and $\partial^{+r}$ denotes the $r$-fold upper shadow. The extremal iterated-shadow conjecture. For every $\epsilon>0$, if $r=\lceil\epsilon\sqrt n\rceil$, then

$$\mu\bigl(\partial^{+r}(\mathcal{A})\cap\partial^{+r}(\mathcal{A}^c)\bigr)=\Omega(\epsilon).$$

The claim proposes that the balanced example based on the threshold $|A\cap[n/2]|>n/4$ is extremal for Friedgut's problem. The source presents it as an open problem and gives no resolution.