Elliptic squares and Zilber–Pink for curves in A2

We prove that every Hodge-generic algebraic curve in ${\mathcal{A}}_2$ over $\overline{\mathbb{Q}}$ contains only finitely many points whose abelian surface is isogenous to the square of an elliptic curve without complex multiplication (CM). Combining this result with the companion $E\times \mathrm{CM}$ and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in ${\mathcal{A}}_2$ over $\overline{\mathbb{Q}}$, without a boundary hypothesis.