Coates–Sujatha fine Selmer conjecture for elliptic curves

Let $K$ be a number field, let $p$ be a prime number, and let $\textbf{K}/K$ be the cyclotomic $\mathbb{Z}_p$-extension. For an elliptic curve $E$ over $K$, let $Y(E/\textbf{K})$ denote the Pontryagin dual of the fine Selmer group of $E$ over $\textbf{K}$.

Coates–Sujatha's conjecture. For every elliptic curve $E$ over a number field $K$, $Y(E/\textbf{K})$ is a finitely generated $\mathbb{Z}_p$-module.

This is an analogue of Iwasawa's conjecture for fine Selmer groups. The source attributes the formulation to Coates and Sujatha and does not give evidence that it has been resolved.