The $\mathfrak{M}_H(G)$-conjecture for Selmer groups over $p$-adic Lie extensions

Let $F_\infty/F$ be a strongly admissible $p$-adic Lie extension, set $G=\operatorname{Gal}(F_\infty/F)$ and $H=\operatorname{Gal}(F_\infty/F^{\mathrm{cyc}})$, and let $X(E/F_\infty)$ be the Pontryagin dual of the Selmer group. Define

$$X_f(E/F_\infty):=X(E/F_\infty)/X(E/F_\infty)[p^\infty].$$

The $\mathfrak{M}_H(G)$-conjecture. The module $X(E/F_\infty)$ lies in $\mathfrak{M}_H(G)$; equivalently, $X_f(E/F_\infty)$ is finitely generated over $\mathbb{Z}_p[[H]]$. This condition enables the algebraic $K$-theoretic construction of characteristic elements; the source gives no resolution status.