The $\mathfrak M_H(G)$-conjecture for supersingular elliptic curves

Let $E$ be the elliptic curve and let $X_{\infty,f}^{\pm}$, $\Lambda(H)$, and the groups and fields denoted by the notation of Section 1 in the source be as defined there. $\mathfrak M_H(G)$-conjecture.

$$X_{\infty,f}^{\pm}\text{ is a finitely generated }\Lambda(H)\text{-module}.$$

This is the supersingular equivalent formulation of the $\mathfrak M_H(G)$-conjecture, previously appearing in the cited work of Lei and Zerbes and in Conjecture 4.14 of Lei and Lim. The source notes a criterion for its validity, but the supplied material does not establish whether the conjecture is resolved.