Three-layer conjecture for maximum antichains in a random subset of the Boolean lattice

Let $\mathcal{P}(n)_p$ be the random subfamily of the Boolean lattice $\mathcal{P}(n)$ obtained by including each subset independently with probability $p$, and let a maximum antichain be an antichain of largest possible size in $\mathcal{P}(n)_p$. Here $p=\omega(1/n)$ means that $pn\to\infty$ as $n\to\infty$. Three-layer conjecture. For $p=\omega(1/n)$, every maximum antichain of $\mathcal{P}(n)_p$ is contained in three consecutive layers, with high probability. The conjecture proposes that above the threshold at which the width is asymptotically equal to the size of a middle layer, maximum antichains are localized near the middle and have only a bounded number of layers of support; the paper proves the corresponding structural description for constant $p>1/2$, while the asserted range $p=\omega(1/n)$ remains open.