Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of degree $n$ with coefficients in $\{-1, 1\}$, $$\max_{|z|=1} |P(z)| > (1 + c) \sqrt{n}?$$

1True ↔2  ∃ c > 0,3    ∀ᶠ (n : ℕ) in Filter.atTop,4      ∀ (P : Polynomial ℂ),5        (∀ i ≤ P.natDegree, P.coeff i = -1 ∨ P.coeff i = 1) →6          P.natDegree = n → ⨆ z, ‖Polynomial.eval (↑z) P‖ > (1 + c) * √↑n