A remark on an integral structure of the imperfect coefficient ring of $(\varphi,\Gamma)$-modules

Let $K$ be a complete discrete valuation field of characteristic $0$ with perfect residue field of characteristic $p>0$. Let $\mathbb{A}_K$ denote the imperfect coefficient ring of $(\varphi,\Gamma)$-modules defined by Jean-Marc Fontaine. We prove that the canonical map $W(k_{K_\infty})[[\mu]]\rightarrow \mathbb{A}_K\cap A_\mathrm{inf}$ is an isomorphism, even if $K$ is ramified. This fact was remarked by Nathalie Wach without proof.