Let $E$ be an elliptic curve defined over $\mathbb{Q}$, let $$K:=\bigcup_{k\in\mathbb{Q}[\mu_{{p}^{\infty}}]}\mathbb{Q}\left[\mu_{{p}^{\infty}},k^{{\frac{{1}}{{p}^{\infty}}}}\right]$$ and $G:=\operatorname{Gal}(\overline{K}/K)$. Suppose that ${E}_{p}(K)=0$.

Question: Is there always a $\tau\in G$ so that $$\dim_{\mathbf{Z}_{p}}(E_{{p}^{\infty}}(\overline{K})/(\tau-1)E_{{p}^{\infty}}(\overline{K}))=1$$