Largest-component conjecture for Boolean sublattices

Let $[n]=\{1,\ldots,n\}$, let $2^{[n]}$ be its power set, and let $G_{\mathcal{F}}$ be the graph whose vertex set is a family $\mathcal{F}\subseteq 2^{[n]}$, with two vertices adjacent exactly when they form a $2$-chain under inclusion. Let $k,n$ be integers with $0\le k\le n$ such that $k=0$, $k=1$ or $k$ has the same parity as $n$. Largest-component conjecture. If the components of $G_{\mathcal{F}}$ have order at most $2^k$, then

$$|\mathcal{F}|\le 2^k\binom{n-k}{\lfloor (n-k)/2\rfloor}.$$

This conjecture asks for the maximum size of a family whose comparability graph has no connected component larger than $2^k$, extending the threshold perspective around Sperner's theorem. Its resolution status is not specified in the supplied text.