Littlewood's flat polynomial conjecture

A Littlewood polynomial is a polynomial with coefficients all equal to $\pm1$. For each nonnegative integer $n$, let $p_n$ be a Littlewood polynomial of degree $n$.

Littlewood's conjecture. There exist positive constants $c_1$ and $c_2$ such that, for every $n$, one can choose $p_n$ satisfying

$$c_1\sqrt{n+1}\leq |p_n(z)|\leq c_2\sqrt{n+1}$$

for every complex number $z$ with $|z|=1$.

The conjecture asks for Littlewood polynomials whose modulus is uniformly comparable to the unavoidable $L_2$ scale $\sqrt{n+1}$ on the unit circle. The source describes this as an older problem that had remained unsolved for more than fifty years.