The infinite-rank conjecture for abelian varieties over finitely generated Galois extensions

Let $A_0$ be a non-trivial abelian variety over a finitely generated extension $K_0$ of $\text{Q}$, and let $\sigma_1,\ldots,\sigma_n\in \operatorname{Gal}(\bar K_0/K_0)$. Write $\bar K_0^{\langle \sigma_1,\ldots,\sigma_n\rangle}$ for the subfield fixed by the subgroup generated by these elements.

The infinite-rank conjecture.

$$\dim_{\mathbb{Q}} A_0(\bar K_0^{\langle \sigma_1,\ldots,\sigma_n\rangle})\otimes \mathbb{Q} = \infty.$$

The paper proves the elliptic-curve case and explains that this conjecture is equivalent to the assertion that every non-trivial abelian variety over a characteristic-zero field with finitely generated Galois group has infinite rank. Earlier work established the assertion in various special cases, but the general conjecture remains open.