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 ${\mathbb{Q}}^{\mathrm{a}\mathrm{l}\mathrm{g}}$ and let $C_1,C_2\subseteq G^N(\mathbb{C})$ be irreducible closed algebraic curves with $N\ge 3$. Further assume that at least one of $C_1$ and $C_2$ is not defined over ${\mathbb{Q}}^{\mathrm{a}\mathrm{l}\mathrm{g}}$. Suppose that there does not exist an algebraic subgroup $G\subseteq G^N(\mathbb{C})$ of dimension $g$ such that $C_1\subseteq G$ and that there does not exist an algebraic subgroup $H\subseteq G^N(\mathbb{C})$ of dimension $2g$ such that $C_1\cup C_2\subseteq H$. Denoting $\mathcal{N}=\{n\in \mathbb{N}\mathrm{\mid }[n]C_1\subseteq C_2\}$, we prove that ${\bigcup }_{n\in \mathbb{N}\setminus \mathcal{N}}\{x\in C_1\mathrm{\mid }{x}^n\in C_2\}$ is finite.