Proof of the Kahn Saks Conjecture

Let $\mathbb{P}(x\prec y)$ be the probability that $x$ precedes $y$ in a uniformly random linear extension of an $n$-element poset $P$, and define the balancing coefficient to be $δ(x,y)=\min(\mathbb{P}(x\prec y),\mathbb{P}(y\prec x))$ with $δ(P)=\max_{x,y}δ(x,y)$. We prove (Theorem 1) that sufficie