Finite generation conjecture for elliptic curves over infinite real abelian extensions

Let $E$ be the elliptic curve under consideration, and let $F/\mathbb{Q}$ be an infinite real abelian extension containing only finitely many subfields of degree $2$, $3$, or $5$ over $\mathbb{Q}$. Infinite-extension finite generation conjecture. The Mordell–Weil group $E(F)$ is finitely generated. This is the infinite-extension arithmetic consequence proposed after the conjecture on vanishing twisted $L$-values; it is open and concerns the scarcity of low-degree subfields in $F$.