Discreteness conjecture for abelian varieties over maximal abelian extensions

Let $K$ be a number field, and let $A/K$ be an abelian variety. The discreteness conjecture for abelian varieties.

1. $A(K^{\rm ab})$ has the discreteness property.
2. If $A$ does not contain an abelian subvariety having complex multiplication over $K$, then $A(K^{\rm ab})$ has the strong discreteness property.

These assertions extend the paper's preceding results for algebraic tori and elliptic curves, while the cited results establish related statements for torsion points and for maximal cyclotomic extensions. The conjecture concerns the corresponding discreteness properties over the maximal abelian extension in general.