Greedy balanced pairs in $N$-free ordered sets

An $alpha$-greedy balanced pair in an ordered set $P=(V,leq)$ is a pair $(x,y)$ of elements of $V$ such that the proportion of greedy linear extensions of $P$ that put $x$ before $y$ among all greedy linear extensions is in the real interval $[alpha, 1-alpha]$. We prove that every $N$-free ordered set which is not totally ordered has a $frac{1}{2}$-greedy balanced pair.