The range conjecture for balancing probabilities in linear extensions

Let $P$ be a finite poset. For $x\in P$, let $\Pi(x)=\{y:y\not\sim x\}$ and define the range parameter

$$\pi(P)=\max_{x\in P}|\Pi(x)|.$$

For distinct $x,y\in P$, let $\delta_{xy}=\min\{\mathbb{P}(x\prec y),\mathbb{P}(y\prec x)\}$, and set

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

where probabilities are taken over uniformly random linear extensions. The range conjecture. If $\pi(P)\to\infty$, then $\delta(P)\to 1/2$. This is presented as an old conjecture of the second author and as a strengthening of the width-based Kahn–Saks conjecture. The supplied text does not state that it has been resolved.