Finite generation conjecture for elliptic curves over real abelian extensions

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $F/\mathbb{Q}$ be a real abelian extension containing only finitely many subfields of degree $2$, $3$, or $5$ over $\mathbb{Q}$. Finite generation conjecture. The group of $F$-rational points $E(F)$ is finitely generated. This is motivated by the paper's heuristic study of diophantine stability and modular-symbol distributions; the conjecture is presented as an open prediction for infinite extensions as well as finite cases.