The infinite-rank conjecture for abelian varieties over fields with finitely generated Galois group

Let $K$ be a field of characteristic zero, and let $G_K=\operatorname{Gal}(\bar K/K)$ be its absolute Galois group. Let $A/K$ be a non-trivial abelian variety; saying that $A$ has infinite rank over $K$ means

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

The infinite-rank conjecture. If $G_K$ is finitely generated, then $A$ has infinite rank over $K$.

This is presented as an equivalent formulation of the preceding conjecture. The paper proves the elliptic-curve case, while the assertion for arbitrary non-trivial abelian varieties and arbitrary characteristic-zero fields with finitely generated Galois group remains open.