The most frequent peak set of a random permutation

Given a subset $S\subseteq\mathbb{P}$, let $\Pa(S;n)$ be the number of permutations in the symmetric group of ${1,2,...,n}$ that have peak set $S$. We prove a recent conjecture due to Billey, Burdzy and Sagan, which determines the sets that maximize $\Pa(S;n)$, where $S$ ranges over all subsets of ${1,2,...,n}$.