Lubin's conjecture for height-one $p$-adic dynamical systems over $p(p-1)$-tame extensions

We prove Lubin's conjecture for height-one commuting pairs of formal power series defined over the ring of integers $\mathcal{O}$ of finite extensions of $\mathbb{Q}_p$ whose ramification index is coprime to $p(p-1)$. For such a pair $(f, u)$, with $f$ noninvertible and $u$ invertible, we show that