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Claim. The answer is no. The question is Problem 7 of the 1958 paper of Erdős, Herzog and Piranian (Section 6): if all the zeros of $f$ lie in the open disc $D_r=\{|z|<r\}$, $r<2$, does the set $E(f)=\{z:|f(z)|<1\}$ have a component of diameter greater than $2-r$? The closed-disc hypothesis $|z|\le r$ of the site's statement is Pommerenke's restatement (p. 98), which also takes $E=\{|f(z)|\le1\}$ closed and asks for a component of diameter at least $2-r$. Pommerenke's unnumbered passage on p. 98, as printed: "The answer is negative for $r>1$. To show this, let $f(z)=z^n-r^n$ ($r>1$). Then the set $E=\{|z^n-r^n|\le1\}$ has $n$ components." Each component contains one zero $r\zeta$, $\zeta^n=1$, and every point $z$ of the component of the zero $r$ satisfies $|z-r|\le n^{-1}r^{-n+1}(1+O(n^{-1}))$, so the common diameter of the components tends to $0$ as $n\to\infty$; for fixed $1<r<2$ and $n$ large no component has diameter $2-r$ or more. Every component of the open set $\{|f|<1\}$ lies in a component of $E$, so the example refutes the question as posed (an observation recorded on the result page, not the paper's sentence). The complementary half is Theorem 3 (p. 99): when the zeros lie in $|z|\le r\le1$, the component $E_0$ of $E$ containing $0$ has diameter $d_0\ge2$ for $0\le r\le1/2$, $d_0>1/r$ for $1/2<r\le(\sqrt5-1)/2$ and $d_0>2-r^2$ for $(\sqrt5-1)/2\le r\le1$, each bound at least $2-r$, so the paper answers Problem 7, in its restated closed form, affirmatively for $0<r\le1$. Two observations on the problem's open set, recorded on this page and not the paper's: for $0<r\le1/2$ the bound carries over, because by the Gauss--Lucas theorem every critical point $w$ of $f$ lies in $|w|\le1/2$, where $|f(w)|\le\prod(|w|+|z_\nu|)\le1$ with equality only for $f=(z+w)^n$, whose critical point is $-w$, so no critical point lies on the level set $|f|=1$, each component of $E$ is the closure of one component of $\{|f|<1\}$, and the open set, connected like $E$, has diameter $d_0\ge2>2-r$; for $1/2<r\le1$ the theorem bounds the closed component $E_0$ and does not decide the strict question for the open set. The statements are on the result pages example_p98 and theorem_3 of the source card pommerenke_1961_metric_properties_complex_polynomials.

Source. Ch. Pommerenke, On metric properties of complex polynomials, Michigan Math. J. 8 (1961), no. 2, 97--115, DOI 10.1307/mmj/1028998561; received November 26, 1960. The publisher's record dates the article to the year 1961 alone, and the page is named by the record's date.

Acceptance. Refereed: the paper appeared in the Michigan Mathematical Journal. Reviewed: the site's curator, T. F. Bloom, labels the problem disproved and credits the negative answer for $r>1$ to this example, with its $n$ components whose diameter tends to $0$, and records the three bounds of Theorem 3 as the affirmative answer for $0<r\le1$. Nothing here is independently reviewed by this project.

Formalization. The Lean file Erdos1048 in Boris Alexeev's repository, linked above at the commit that added it, declares itself a formalization of a solution found by Pommerenke: the statement of Pommerenke's result was given to Aristotle, the system of Harmonic, which formalized the proof. Its main_result proves, for $r>1$, that $\{|z^n-r^n|\le1\}$ has exactly $n$ connected components for every $n\ge1$ and that their diameters are eventually below any $\varepsilon>0$, and not_erdos_1048 refutes a statement erdos_1048 in which the diameter bound reads $2-r\le\operatorname{diam}$. The author's thread post explains the change from the strict inequality: with the strict form and $r=0$ allowed, the polynomial $z^2$, whose sublevel set is the unit disc of diameter exactly $2$, is already a counterexample. Refuting the non-strict form refutes the strict one. The qualifier of the site's label DISPROVED (LEAN) and the formal_proof attribute of the formal-conjectures statement file refer to this development. This corpus has not built or audited it, so no formalized evidence is listed. Aristotle's separate disproof from the problem statement alone, with $z^{10}-2$, is recorded on Alexeev's page.

Depends on. Nothing on the wiki. The example uses the binomial expansion of $(r^n+\omega)^{1/n}$; Theorem 3 uses Lemmas 1 and 2 of the paper and the fact that a continuum of capacity $1$ has diameter at least $2$.