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

Let $K\mathrm{/}{\mathbb{Q}}_p$ be a finite extension whose ramification index is coprime to $p^2-p$. We study height-one commuting pairs $(f,u)$ of noninvertible and invertible formal power series defined over the ring of integers ${\mathcal{O}}_K$ of $K$. We begin by extracting a crystalline character of weight $1$ from the $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{K}\mathrm{/}K)$-set $T_f$ of $f$-consistent sequences. This character is used in order to equip $T_f$ with a ${\mathbb{Z}}_p$-module structure for which $f$ is an endomorphism. We then apply explicit functors in integral $p$-adic Hodge theory to $T_f$ to recover a formal group defined over ${\mathcal{O}}_K$ for which $(f,u)$ is a pair of endomorphisms. This proves new cases of a conjecture of Lubin.