Lubin's conjecture for height-one p-adic dynamical systems

Let $f$ and $u$ be, respectively, a noninvertible and a nontorsion invertible power series over the ring of integers $\mathcal{O}$ of a finite extension of $\mathbb{Q}_p$. Suppose that all roots of $f$ and of its iterates are simple, and that $f'(0)$ is a uniformizer in $\mathbb{Z}_p$. If

$$f \circ u=u \circ f,$$

then $f$ and $u$ are endomorphisms of some formal group $F$ over $\mathcal{O}$.

Lubin's conjecture. Under these hypotheses,

$$f,u\in\operatorname{End}_{\mathcal{O}}(F)$$

for some formal group $F$ over $\mathcal{O}$.

The conjecture formalizes the expectation that a commuting noninvertible and nontorsion invertible pair of $p$-adic power series arises from a formal group. The paper states that it proves the conjecture for height-one commuting pairs over the ring of integers of any finite extension of $\mathbb{Q}_p$, so the conjecture is resolved in the setting described here.