Let $f(z)={\sum }_{0\le k\le n}{ϵ}_k{z}^k$ be a random polynomial, where ${ϵ}_k\in \{-1,1\}$ independently uniformly at random for $0\le k\le n$. Does there exist some constant $C>0$ such that, almost surely,$$\underset{|z|=1}{\max }\left|\sum _{k\le n}{ϵ}_k(t)z^k\right|=(C+o(1))\sqrt{n\log n}?$$