Kahn–Saks conjecture on balancing probabilities in linear extensions

Let $P$ be a finite poset, let $w(P)$ denote its width, and for distinct elements $x,y\in P$ define

$$\delta_{xy}=\min\{\mathbb{P}(x\prec y),\mathbb{P}(y\prec x)\},$$

with

$$\delta(P)=\max_{x\ne y}\delta_{xy},$$

where the probabilities are over a uniformly random linear extension of $P$. Kahn–Saks conjecture. If $w(P)\to\infty$, then $\delta(P)\to 1/2$. The conjecture formalizes the expectation that posets of increasingly large width contain a pair of elements whose relative order is asymptotically balanced. Its status is not resolved in the supplied text.