Erdős Problem #228 — Flat Littlewood Polynomials

Do there exist constants $c_1,c_2\in\mathbb{R}$ such that, for all sufficiently large integers $n$, there is a polynomial $p\in\mathbb{C}[z]$ of degree $n$ whose coefficients satisfy $[z^i]p\in\{1,-1\}$ for every integer $0\leq i\leq n$, and such that for every $z\in\mathbb{C}$ with $|z|=1$,

$$\sqrt{n}<c_1|p(z)|\quad\text{and}\quad |p(z)|<c_2\sqrt{n}?$$