The maximal length of the Erdős--Herzog--Piranian lemniscate in high degree

Let $n\ge 1$, and let $p:\mathbf{C}\to \mathbf{C}$ be a monic polynomial of degree $n$. It was conjectured by Erdős, Herzog, and Piranian that the maximal length of lemniscate $\{z\in \mathbf{C}:\mathrm{\mid }p(z)\mathrm{\mid }=1\}$ is attained by the polynomial $p(z)=z^n-1$. In this paper, building upon a previous analysis of Fryntov and Nazarov, we establish this conjecture for all sufficiently large $n$.