Points and their multiples on curves in powers of simple abelian varieties

Let $G$ be a simple abelian variety of dimension $g \in \mathbb{N}$ defined over $\mathbb{Q}^\mathrm{alg}$ and let $C_1, C_2 \subseteq G^N(\mathbb{C})$ be irreducible closed algebraic curves with $N \geq 3$. Further assume that at least one of $C_1$ and $C_2$ is not defined over $\mathbb{Q}^\mathrm{