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Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $\pm 1$, such that \[\sqrt{n} \ll \lvert P(z) \rvert \ll \sqrt{n}\] for all $\lvert z\rvert =1$, with the implied constants independent of $z$ and $n$?
Originally a conjecture of Littlewood. The answer is yes (for all $n\geq 2$), proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST19].

See also [230].

Additional thanks to: Mehtaab Sawhney