If $p(z)$ is a polynomial of degree $n$ such that $\{z : \lvert p(z)\rvert\leq 1\}$ is connected then is it true that $$\max_{\substack{z\in\mathbb{C}\\ \lvert p(z)\rvert\leq 1}} \lvert p'(z)\rvert \leq (\tfrac{1}{2}+o(1))n^2?$$

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1True ↔2  ∀ ε > 0,3    ∀ᶠ (n : ℕ) in Filter.atTop,4      ∀ (p : Polynomial ℂ),5        p.Monic →6          p.natDegree = n →7            IsConnected {z | ‖Polynomial.eval z p‖ ≤ 1} →8              ∀ (z : ℂ),9                ‖Polynomial.eval z p‖ ≤ 1 → ‖Polynomial.eval z (Polynomial.derivative p)‖ ≤ (1 / 2 + ε) * ↑n ^ 2

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