The site labels the problem DISPROVED, but its commentary answers the stated question yes: "The answer is yes, and in fact the centre of this disc can be taken to be $\frac{z_1+\cdots+z_n}{n}$, where the $z_i$ are the roots of $f$, as shown by Pommerenke [Po59]." Pommerenke's Theorem 3 (Michigan Math. J. 6 (1959), p. 222) states that if $E$ is connected then the lemniscate $|f(z)|=1$ lies in the circle $|z-\zeta|<2$, $\zeta$ the centroid of the zeros, and $E$, bounded by that lemniscate, lies in the same open disc; the paper introduces the theorem as establishing "the conjecture in Problem 14" of [EHP58], the question stated above. So the Statement is proved (Pommerenke's claim page (1959), accepted, full, refereed). The only refutation the commentary reports is of a different conjecture from the same passage of [EHP58], Problem 15, that the width of a connected $\{|f|\le1\}$ is at most $2$: the Remarks of the same paper (pp. 224--225) give $\sup b\ge\sqrt3\,2^{1/3}>2.18$. The site's label fits that conjecture, not the Statement, and the commentary names nothing else as false. The page departs from the site's label here: DISPROVED, the site's "solved in the negative", contradicts Pommerenke's theorem in print and the site's own commentary, and the formal-conjectures statement for the problem is tagged solved with its proof attribute pointing to a Lean proof of the affirmative.