Larsen's infinite-rank conjecture for abelian varieties over algebraic extensions

Let $K/\mathbb{Q}$ be an algebraic extension with finitely generated absolute Galois group $G_K$, and let $A$ be a non-zero abelian variety over $K$. Define

$$\operatorname{rank}(A(K))=\dim_{\mathbb{Q}}(A(K)\otimes\mathbb{Q}).$$

Larsen's conjecture. Then

$$\operatorname{rank}(A(K))=\infty.$$

A positive answer to the question whether every infinite field with finitely generated absolute Galois group is ample would settle this conjecture, using the infinite-rank result for non-zero abelian varieties over ample fields of characteristic zero. Its general status is open.