The 1/3-2/3 conjecture for $N$-free ordered sets

A balanced pair in a finite ordered set $P=(V,\le )$ is a pair $(x,y)$ of elements of $V$ such that the proportion of linear extensions of $P$ that put $x$ before $y$ is in the real interval $[1\mathrm{/}3,2\mathrm{/}3]$. We prove that every finite $N$-free ordered set which is not totally ordered has a balanced pair.