Sparse Sperner conjecture for three central layers

Let $B_n$ be the Boolean lattice on $[n]$, let $N$ denote the size of a largest layer of $B_n$, and call the layers closest to the middle the central layers. For a function $\beta=\beta(n)$, consider antichains of size $\beta N$ in $B_n$.

Sparse Sperner conjecture. There exists a function $\beta^*(n)=\tilde\Theta(1/n)$ such that if $\beta\geq\beta^*(n)$, then almost all antichains of size $\beta N$ in $B_n$ are contained in three central layers.

This strengthens the proved result that the same conclusion holds for $\beta>C\log^2 n/\sqrt n$ for an absolute constant $C$. The conjectured threshold is motivated by the expected appearance of vertices two layers outside the middle, and the claim would follow from a suitable extension of the relevant container result to $\lambda=\tilde\Omega(1/n)$.