The one-third–two-thirds conjecture for finite posets

Let $P=(X,\prec)$ be a finite poset, let $\operatorname{\mathcal{E}}(P)$ be its set of linear extensions, and write $e(P)=|\operatorname{\mathcal{E}}(P)|$. The one-third–two-thirds conjecture. If $P$ is not totally ordered, then there are distinct elements $x,y\in X$ such that

$$\frac13\leq \frac{|\\{L\in\operatorname{\mathcal{E}}(P):L(x)<L(y)\\}|}{e(P)}\leq\frac23.$$

This conjecture is open in full generality, although it is known for several special classes, including posets of width two, where the bounds are tight.