  Let $E/\mathbb{Q}$ be an elliptic curve, $p$ an odd prime and $K_{\infty}/K$
the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$.
In a previous article the author conjectured that the fine $p^{\infty}$-Selmer
group $R_{p^{\infty}}(E/K_{\infty})$ is confinitely generated over
$\mathbb{Z}_p$. In this note we prove this conjecture assuming some hypotheses
on $E$ and $p$.
