Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}$, maximal ideal $\mathfrak{m}$ and uniformizer $\pi$. Let $\bar K$ be the algebraic closure of $K$ and $\bar{\mathfrak{m}}$ be the integral closure of $\mathfrak{m}$.

Let $f(x) \in x \mathcal{O}[[x]]$ be a noninvertible power series with finite Weirstrass-degree and denote $f^{\circ n}(x)$ be the $n^{th}$ iterates of $f(x)$. We denote the set of solutions of $f^{\circ n}(x)$ by $S_n=\{x \in \bar{\mathfrak{m}}~|~f^{\circ n}(x)=0 \}$. Also denote $S=\bigcup_n S_n$.

Since $f$ has finite Weierstrass degree, $\#S_n$ is finite. Assume that the field extension $K(S_n)$ be Galois such that \begin{align}\text{Gal}(K(S_n)/K) \cong \left( \mathcal{O}/\pi^n \mathcal{O} \right)^{\times},~~~~~~~~~~~(1)\end{align} Taking inverse limit of the relation $(1)$, we get \begin{align} \text{Gal}(\bar K/K) \cong \mathcal{O}^{\times},~~~~~~~~~~~~(2) \end{align} There is a continuous surjection $\text{Gal}(\bar K/K) \twoheadrightarrow \text{Gal}(K(S_n)/K)$ by $\sigma \mapsto \sigma|_{K(S_n)}$.

My question-

Is the absolute Galois group $\text{Gal}(\bar K/K)$ isomorphic to $\text{Gal}(K(S)/K)$ ?

Thanks