Infinite-rank conjecture for elliptic curves over fields with finitely generated Galois group

Let $K$) be a field that is not locally finite, and let

$$G_K=\operatorname{Gal}(K^{\operatorname{sep}}/K)$$

be topologically finitely generated. Let $E/K$ be a non-trivial elliptic curve. Infinite-rank conjecture. The elliptic curve $E$ has infinite rank over $K$. This conjecture is motivated by results proving the assertion under stronger hypotheses on the field or the elliptic curve; the general case is presented as open here.