An inclusion coupling of prefix laws and the Chan-Pak rank inequality for all finite posets

Let $P$ be a finite poset, let $x\in P$, and let $N_k$ count its linear extensions in which $x$ occupies position $k$. We prove Chan and Pak's conjecture that, for all integers $a,h\ge1$, \[ (a+h-1)N_{a+h-1}N_a\ \ge\ (a-1)N_{a-1}N_{a+h}+hN_aN_{a+h}. \] The key result is an inclusion coupling of the