Mazur's conjecture for abelian varieties

Let $A$ be an abelian variety over ${\mathbb Q}$, and let $G$ be a subgroup of $A({\mathbb Q})$. For a certain abelian subvariety $B$ of $A$ defined over ${\mathbb Q}$, the closure of $G$ in the Euclidean topology of $A({\mathbb R})$ contains $B({\mathbb R})^0$ as a subgroup of finite index.

Mazur's conjecture for abelian varieties. The preceding assertion should hold for every such $A$ and $G$.

Mazur proposed this as a density statement for rational points. The source notes that the general conjecture is false, while a reformulated version for abelian varieties is presented as supported by the available evidence; Waldschmidt's theorem gives a partial result for simple abelian varieties under a sufficiently large Mordell–Weil rank.