The 1/3-2/3 conjecture for linear extensions

Let $P$ be a finite poset. For distinct elements $x,y\in P$, define

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

and let

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

where the probabilities are taken over a uniformly random linear extension of $P$. The 1/3-2/3 conjecture. If $P$ is not a chain, then $\delta(P)\geq 1/3$. This conjecture is motivated by sorting problems; the bound is known to be tight, but the conjectured constant $1/3$ remains unresolved, with the best stated general lower bound being $(5-\sqrt{5})/10$.