Frankl–Akiyama conjecture on antichains in convex families

Let $\mathcal{B}_n$ be the Boolean lattice of subsets of $[n]$. For a family $\mathcal{F}\subseteq\mathcal{B}_n$, define its convex closure by

$$\overline{\mathcal{F}}=\{S\in\mathcal{B}_n\mid A\subseteq S\subseteq B\text{ for some }A,B\in\mathcal{F}\}.$$

A family is convex when $\mathcal{F}=\overline{\mathcal{F}}$, and an antichain is a family of pairwise incomparable sets. Frankl–Akiyama conjecture. For every convex family $\mathcal{F}\subseteq\mathcal{B}_n$, there exists an antichain $\mathcal{A}\subseteq\mathcal{F}$ such that

$$\frac{|\mathcal{A}|}{|\mathcal{F}|}\geq\frac{\binom{n}{\lfloor n/2\rfloor}}{2^n}.$$

The conjecture asserts that every convex subfamily of the Boolean lattice contains an antichain whose relative size is at least the maximum rank proportion in the whole Boolean lattice. Its status is not resolved by the supplied source context.