Erdős Problem #230 — Unimodular polynomials bounded away from the Parseval limit

Is there an absolute constant $c>0$ such that every $n≥2$ and every polynomial $P(z)=∑_{k=1}^n a_kz^k$ with $|a_k|=1$ satisfy $\max_{|z|=1}|P(z)|≥(1+c)√n$?