Selmer groups and anticyclotomic $\mathbb{Z}_p$-extensions II

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime where $E$ has ordinary reduction and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. We give sufficient conditions on $E$ and $p$ in order to ensure that $\text{Sel}_{p^{\infty}}(E/K_{\infty})$ is a cofree $\Lambda$-module of rank one. We also show that these conditions imply that $\text{rank}(E(K_n))=p^n$ for all $n \geq 0$ and that the $p$-primary subgroup of the Tate-Shafarevich group of $E/K_n$ is trivial for all $n \geq 0$.