If $f\in \mathbb{C}[x]$ is a monic polynomial with all roots satisfying $\lvert z\rvert \leq r$ for some $r<2$, then must $$\{ z: \lvert f(z)\rvert <1\}$$ have a connected component with diameter $>2-r$?

1∀ (r : ℝ),2  0 < r →3    r ≤ 1 →4      ∀ (f : Polynomial ℂ),5        f.Monic →6          f.degree ≥ 1 →7            (∀ z ∈ f.roots, ‖z‖ ≤ r) →8              ∃ z ∈ Erdos1048.openLevelSet f,9                ENNReal.ofReal (2 - r) < Metric.ediam (connectedComponentIn (Erdos1048.openLevelSet f) z)