Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $\pm1$, such that $$\sqrt n \ll |P(z)| \ll \sqrt n$$ for all $|z|=1$, with the implied constants independent of $z$ and $n$?

1True ↔2  ∃ c₁ c₂,3    ∀ᶠ (n : ℕ) in Filter.atTop,4      ∃ p,5        p.degree = ↑n ∧6          (∀ i ≤ n, p.coeff i = 1 ∨ p.coeff i = -1) ∧7            ∀ (z : ℂ), ‖z‖ = 1 → √↑n < c₁ * ‖Polynomial.eval z p‖ ∧ ‖Polynomial.eval z p‖ < c₂ * √↑n