Erdős Problem #523 — The asymptotic maximum of a random sign polynomial

Let $ε_0,ε_1,…$ be independent uniform random signs. Is there a constant $C>0$ such that, almost surely, $\max_{|z|=1}|∑_{k=0}^n ε_kz^k|=(C+o(1))√{n\log n}$ as $n→∞$?