Generalised Shafarevich conjecture for Shimura-type integral structures

Let $K=K^lK_l$ be a neat compact open subgroup of $G(\mathbb A_f)$, let $x=[s,\overline{1}]$ be a point of $Sh_K(G,X)(L)$, and fix a lift $\widetilde{x}=[s,1]$ in $Sh_{K^l}(G,X)$. Let $(V_{\mathbb Z},s,\rho_{\widetilde{x},l})$ be the corresponding $(\infty,l)$-integral structure of Shimura type. Generalised Shafarevich conjecture. The set of $(\infty,l)$-integral structures of Shimura type that are $l$-isogeneous to $(V_{\mathbb Z},s,\rho_{\widetilde{x},l})$ contains only finitely many isomorphism classes. The paper studies this finiteness assertion and its relation to the Isogeny conjecture; the source does not state that it has been resolved.