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Claim. The answer is no. There is a monic polynomial $f$ such that the projection of $\{z:|f(z)|\le1\}$ onto every straight line has measure greater than $2.386$. Pommerenke obtains it on p. 103 of the 1961 paper by applying the approximation theorem of its p. 97 (a closed bounded set of capacity $1$ lies in the interior of a lemniscate curve $\{|f|=\rho^n\}$, with $f$ monic and $\rho$ slightly above $1$, and the curve lies in an $\varepsilon$-neighborhood of the set) to the five-armed star of capacity $1$ from his Math. Ann. 139 paper, whose least projection exceeds $2.386$. The same page bounds the other side: by Theorem 7, every closed bounded set of capacity $1$, so every lemniscate set, has some projection of measure below $3.30$, and by Theorem 6 (p. 102) every projection of a degree-$n$ lemniscate set has measure at most $4\cdot2^{-1/n}$. The paper prints neither the construction of the five-armed star nor the derivation of the bound $2.386$, citing its Math. Ann. 139 paper for both. The passage and Theorem 7 are on the result page example_p103 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 to this paper, adding, under its key [Po59], that the 1961 answer uses Pommerenke's previous work. The commentary answers a thread post of 12 October 2025 that names that work as Pommerenke's Über die Kapazität ebener Kontinuen, Math. Ann. 139 (1959), 64--75, the source of the five-armed star the 1961 paper uses on p. 103. The site's reference record resolves [Po59] to the 1959 Michigan note listed on the problem page, which does not treat the projection question; the 1961 paper cites that note for Problem 10b (Theorem 2, p. 98) and for the width and diameter bounds of p. 109, not for the example of p. 103. Nothing here is independently reviewed by this project.

Lean. The Lean qualifier of the site's label refers to a Lean disproof by a different construction, $z^{16}-1$, built and audited by this corpus and recorded as its own accepted claim on Alexeev's page; no Lean development formalizes this paper's construction.

Depends on. Nothing on the wiki. The result rests on the cited paper and on the five-armed star of Pommerenke's Math. Ann. 139 paper, which is not held.