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Claim. The answer is no, in the problem's exact terms. Theorem 14 of Pommerenke's 1961 paper, as printed on p. 112: "Let $f(z)=z^p(z-a)$. If $a-(1+p^{-1})\cdot p^{1/(p+1)}$ is positive and sufficiently small, and $p$ is sufficiently large, then the set $E=\{|f(z)|\le1\}$ has two components, one of which is not convex." Here $f$ is monic with $m=2$ distinct roots, $0$ and $a$, the level is $c=1$, the closed sublevel set is the problem's, it has $m=2$ components, and the component containing $0$ is not convex. The paper introduces the theorem (pp. 111--112) as a counterexample to the question that Grunsky raised and that Erdős, Herzog and Piranian reported as their Problem 16. The proof (p. 112) is one paragraph: at $a_0=(1+p^{-1})\,p^{1/(p+1)}$ the point $\xi=p^{1/(p+1)}$ is a double point of the level curve, with tangents $y=\pm(x-\xi)$, near which the set lies in the double sector $|y|\le1.1\,|\xi-x|$; the point $0.5+0.6i$ lies in the set for large $p$ but outside that sector, so the segment from it to $\xi$ leaves the set; increasing $a$ slightly separates the two parts and keeps the component of $0$ nonconvex. The statement is on the result page theorem_14 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 theorem, stating the polynomial and the parameter range. Nothing here is independently reviewed by this project.

Other disproofs. Goodman's 1966 quartics, one with four simple roots, disprove Grunsky's question for the open sublevel set at a critical level, and the paper does not treat the problem's closed set (Goodman's page); the Lean qualifier of the site's label refers to a Lean proof with $z^6-z$, built and checked by this corpus's verification and accepted on Alexeev's page.

Depends on. Nothing on the wiki; the proof uses elementary calculus only.