The perfectoid commutant of Lubin-Tate power series

Let LT be a Lubin-Tate formal group attached to a finite extension of Qp. By a theorem of Lubin-Sarkis, an invertible characteristic p power series that commutes with the elements of Aut(LT) is itself in Aut(LT). We extend this result to perfectoid power series, by lifting such a power series to characteristic zero and using the theory of locally analytic vectors in certain rings of p-adic periods. This allows us to recover the field of norms of the Lubin-Tate extension from its completed perfection.