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Claim. Grunsky's question has a negative answer already in degree $4$, and with simple roots. Goodman works with the open set $E(c)=\{z:|P(z)|<c\}$ and its boundary lemniscate. First example (p. 359): for $P(z)=(z^2+1)(z-2)^2$ and $c=|P((1\pm i)/2)|=5\sqrt5/4$, the lemniscate has double points at the two critical points $(1\pm i)/2$, $E(c)$ has three components, and the component containing $2$ has both double points on its boundary but omits their midpoint $1/2$, where $|P(1/2)|=45/16>c$, so it is not convex. Theorem (p. 361): for $Q(z)=3z^4-20z^3+6z^2-60z+303$ and $c=\sqrt{91{,}600}$, the roots of $Q$ are simple and one of the four components of $E(c)$ fails to be convex; dividing $Q$ and $c$ by $3$ gives the monic form. The paper also records the referee's example $P(z)=z(z^5-1)$ with $c=5/6^{6/5}$, whose lemniscate has five double points on the boundary of the component of $0$, and raises the question of the greatest number of nonconvex components as a function of the degree. The statements are on the source card goodman_1966_convexity_level_curves_polynomial.

Closed set. The problem is posed for the closed set $\{z:|f(z)|\le c\}$ with as many components as distinct roots. In both of Goodman's examples $c$ is a critical value at which the lemniscate has double points, so the closed set at that same $c$ has fewer components than the open set, and the paper does not treat the closed set. The disproof in the problem's exact terms is Pommerenke's Theorem 14.

Source. A. W. Goodman, On the convexity of the level curves of a polynomial, Proc. Amer. Math. Soc. 17 (1966), no. 2, 358--361, DOI 10.1090/S0002-9939-1966-0188408-3; received by the editors August 12, 1965, presented to the Society January 24, 1966. The publisher's record dates the issue to April 1966 without a day, and the page is named by the first of that month.

Acceptance. Refereed: the paper appeared in the Proceedings of the American Mathematical Society. Reviewed: the site's curator, T. F. Bloom, labels the problem disproved and credits Goodman's first example, the simple-roots quartic and the referee's example in the problem's commentary. Nothing here is independently reviewed by this project.

Depends on. Nothing on the wiki; the examples are verified by elementary computation in the paper.