Mazur's finite generation conjecture over cyclotomic extensions

Let $\ell$ be a rational prime, let $K$ be a number field, and let $K_\infty=K\cdot\mathbb{Q}_{\infty,\ell}$ be the cyclotomic $\mathbb{Z}_\ell$-extension of $K$. Let $A/K_\infty$ be an abelian variety. Mazur's conjecture. The group $A(K_\infty)$ is finitely generated. This conjecture asks whether the Mordell–Weil theorem extends from number fields to cyclotomic $\mathbb{Z}_\ell$-extensions. It is known for certain elliptic curves, including elliptic curves over $\mathbb{Q}$, but remains open in general.