  Let $\varphi_p(z)=(z-1)^p+2-\zeta_p$, where $\zeta_p\in\bar{\mathbb{Q}}$ is a
primitive $p$-th root of unity for some odd prime $p$. Building on previous
work, we show that the $n$-th iterate $\varphi_p^n(z)$ has Galois group
$[C_p]^n$, an iterated wreath product of cyclic groups, whenever $p$ is not a
Wieferich prime.
