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

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