The site's label is DISPROVED (LEAN). Pommerenke's 1961 example $z^n-r^n$, whose sublevel set for $1<r<2$ has $n$ components of diameter tending to $0$, is the accepted disproof, refuting the question for every $1<r<2$ (Pommerenke's claim page (1961)). The same paper's Theorem 3 gives the affirmative answer for $0<r\le1$ for the closed set $\{|f|\le1\}$ of Pommerenke's restatement; for $0<r\le1/2$ the bound carries over to the problem's open set, while for $1/2<r\le1$ the paper bounds the closed set only (both observations are recorded on the claim page). The degenerate case $r=0$ fails the strict inequality: $z^n$ has the open unit disc, of diameter exactly $2$, as its sublevel set. The label's qualifier (LEAN) refers to Alexeev's formalization of Pommerenke's example, which its author reports as verified by Lean, written by Aristotle and linked from that page. Aristotle's separate disproof with $z^{10}-2$, published in Lean by Alexeev, is a second accepted disproof: this corpus built and audited a later revision of its development (Alexeev's claim page (2026)).