General validity of the disk inequality \(|(z)_n|\geq n|z|\)

Determine whether the inequality \(|1+z+\cdots+z^{n-1}|\geq n|z|\) holds for every positive integer \(n\) and every complex number \(z\) in the closed disk \(\{|z-2|\leq 1\}\), beyond the established cases \(n=5,6,7\).

Background

The paper proves the inequality $|(z)_5|\geq5|z|$ and $|(z)_6|\geq6|z|$ on the disk $\mathcal B=\{|z-2|\leq1\}$. The authors note that analogous numerical evidence suggests the inequality may also hold for $n=7$, while the minimum normalized value appears to approach $2\pi$ as $n$ grows.

This leaves unresolved whether the bound holds for other indices. The statement is included because the authors explicitly report that they do not have a proof and distinguish the known small cases from the conjectural general behavior.