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Claim. The answer is yes, with the center at the centroid of the zeros. Theorem 3 of Pommerenke's 1959 note, as printed on p. 222 with $C$ the lemniscate $|f(z)|=1$ and $E$ its interior $|f(z)|<1$: "Let $\zeta=(z_1+\cdots+z_n)/n$, where $z_1,\cdots,z_n$ are the zeros of $f(z)$. If $E$ is connected, then $C$ is contained in the circle $|z-\zeta|<2$." Since $E$ is bounded and its boundary lies on $C$, $E$ lies in the same open disc (an elementary remark, not the paper's sentence). The paper introduces the theorem as establishing the conjecture in Problem 14 of the 1958 paper of Erdős, Herzog and Piranian, which asks whether $E$ lies in a disc of radius $2$ and whether the center can be the centroid of the zeros; both parts are answered yes, with the strict inequality. The proof (pp. 222--223) applies a Pólya--Szegő bound on the image of the unit circle under a map $w+\zeta+\cdots$ univalent outside the unit disc, that map being the inverse of $f^{1/n}$, univalent because $E$ is connected. The statement is on the result page theorem_3 of the source card pommerenke_1959_some_problems_erdos_herzog_piranian.

Source. Chr. Pommerenke, On some problems by Erdös, Herzog and Piranian, Michigan Math. J. 6 (1959), no. 3, 221--225, DOI 10.1307/mmj/1028998227; received January 15, 1959. The publisher's record dates the article to the year 1959 alone, and the page is named by the record's date.

Acceptance. Refereed: the note appeared in the Michigan Mathematical Journal. The site's commentary, by its curator T. F. Bloom, records the answer to the stated question as yes, with the center of the disc at the centroid of the roots, and credits this paper. The site labels the problem DISPROVED, which contradicts that answer and names no result, so the curator's credit is not counted as review, and the page departs from the site's label for that reason. The only refutation the commentary reports is of a different conjecture from the same 1958 passage, Problem 15 of the 1958 paper, that the width of a connected $E$ is at most $2$: the Remarks of Pommerenke's paper (pp. 224--225) compute the width $\sqrt3\,2^{1/3}$ exactly for a three-segment set and deduce $\sup b\ge\sqrt3\,2^{1/3}>2.18$ for the width $b$ over the class with $E$ connected, recorded on the problem page's reference entry and on the card's theorem_4 page. The claim value here states the mathematical outcome for the stated question. Nothing here is independently reviewed by this project.

Formalization. Boris Alexeev's repository of Lean proofs holds a file Erdos1046.lean, added on 2026-08-17 and linked above at the commit the formal-conjectures file cites, which declares itself a Lean formalization of a solution to the problem, names Pommerenke as its informal author and the AI systems Codex and GPT-5.6 Sol as its formal authors, and proves Erdos1046.erdos_1046, the inclusion of the connected open lemniscate of a monic polynomial in the open disc of radius $2$ about the centroid of the roots, through pommerenke_centroid_bound. The formal-conjectures statement file for the problem cites this file under its formal_proof attributes, as the problem page records. It is third-party Lean that this corpus has not built or audited, so no formalized evidence is listed.

Depends on. Nothing on the wiki. The proof rests on the cited paper and on a Pólya--Szegő problem (Aufgaben und Lehrsätze, Vol. 2, Section IV, Problem 140), which is not held.