The one-third–two-thirds conjecture for sorting probability

Let $P=(X,\prec)$ be a finite poset, and let $\delta(P)$ denote its sorting probability. A poset is a chain if every pair of elements is comparable.

The one-third–two-thirds conjecture. For every finite poset $P=(X,\prec)$ that is not a chain, we have

$$\delta(P)\le \frac13.$$

This conjecture is known for several classes of posets, including Young diagrams and skew Young diagrams, but remains open for general finite posets. The bound is tight for the three-element poset consisting of a two-element chain and an isolated element.