Linear extensions of two-dimensional posets

Let $P$ be a two-dimensional poset on $n$ elements and let $\overline{P}$ be its complement. An inequality due originally to Sidorenko says that $e(P)e(\overline{P})\geq n!$ where $e(P)$ denotes the number of linear extensions of $P$. In this talk, I will discuss proofs to this inequality, one due to Sidorenko and another one, which is geometric, due to Bollobás, Brightwell and Sidorenko. Time permitted, I will talk about the first combinatorial proof, joint with Christian Gaetz, expanding on what I won't have time for during the Friday seminar talk.