The Kahn–Saks conjecture on balance in wide posets

Let $P$ be a finite poset, let $w(P)$ denote its width, and let $\delta(P)$ be the maximum, over distinct $x,y\in P$, of $\min\\{p(x\prec y),p(y\prec x)\\}$. The Kahn–Saks conjecture. If the width tends to infinity, then

$$w(P)\longrightarrow\infty \quad\Longrightarrow\quad \delta(P)\longrightarrow \frac12.$$

The conjecture predicts asymptotically optimal balance for posets of growing width. The text identifies it as still open; it is related to several stronger-looking parameter conjectures developed later in the paper.