The $\mathfrak{M}_H(G)$-conjecture for arithmetic Iwasawa modules

Let $F_\infty/F$ be a $p$-adic Lie extension containing $F^{\mathrm{cyc}}$, with $G={\mathrm{Gal}}(F_\infty/F)$ and $H={\mathrm{Gal}}(F_\infty/F^{\mathrm{cyc}})$. For a finitely generated $\Lambda(G)$-module $M$, let $M(p)$ denote its ${\mathbb Z}_p$-torsion submodule, and define $\mathfrak{M}_H(G)$ to be the category of finitely generated $\Lambda(G)$-modules for which $M/M(p)$ is finitely generated over $\Lambda(H)$. Let $E$ be as above and define

$$X(E/F_\infty):={\mathrm{Hom}}_{\rm cts}({\mathrm{Sel}}(E/F_\infty),{\mathbb {Q}}_p/{\mathbb {Z}}_p).$$

$\mathfrak{M}_H(G)$-conjecture. The Iwasawa module $X(E/F_\infty)$ belongs to the category $\mathfrak{M}_H(G)$. This is the non-commutative Iwasawa-theoretic expectation that arithmetic torsion $\Lambda(G)$-modules satisfy the indicated finiteness condition; the source gives no general resolution.