The Coates--Sujatha module conjecture for false Tate curve extensions

Let $p$ be an odd prime, let $F_\infty=\mathbb{Q}(\mu_{p^\infty},\sqrt[p^n]{m}:n\geq1)$, let $G=\operatorname{Gal}(F_\infty/\mathbb{Q})$, and let $H=\operatorname{Gal}(F_\infty/\mathbb{Q}^{\mathrm{cyc}})$. Define $\mathfrak M_H(G)$ to be the category of finitely generated $\Lambda(G)$-modules $M$ for which $M/M[p^\infty]$ is finitely generated over $\Lambda(H)$. Let $X(E/F_\infty)$ be the Pontryagin dual of the $p^\infty$-Selmer group. Coates--Sujatha module conjecture.

$$X(E/F_\infty)\in\mathfrak M_H(G).$$

This strengthens mere $\Lambda(G)$-torsionness and is a structural conjecture in non-commutative Iwasawa theory. The source gives the statement in an appendix but does not report a resolution.