Fix an abelian variety $A$ of dimension at least $3$ and a (smooth, projective, irreducible) curve $C$ inside $A$ that generates $A$ as an algebraic group, over an algebraically closed field. Now, fix a curve $X$ (also smooth, projective, irreducible) and consider the set $S$ of morphisms $f:X \to A$ with $f(X) \ne C$. Is $\# (f(X) \cap C)$ bounded or unbounded as $f$ varies in $S$?