Larsen's conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$

We prove Larsen’s conjecture for elliptic curves over $\mathbb{Q}$ with analytic rank at most $1$. Specifically, let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$. If $E\mathrm{/}\mathbb{Q}$ has analytic rank at most $1$, then we prove that for any topologically finitely generated subgroup $G$ of $\mathrm{G}\mathrm{a}\mathrm{l}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})$, the rank of $E$ over the fixed subfield ${\bar{\mathbb{Q}}}^G$ of $\bar{\mathbb{Q}}$ under $G$ is infinite.