I've been reading the following Paper (https://arxiv.org/abs/1809.06818) on arithmetic hyperbolicity. We say that a scheme $X/\overline{\mathbb{Q}}$ is arithmetically hyperbolic over $\overline{\mathbb{Q}}$ is for every $\mathbb{Z}$-finitely generated subring $A\subset \overline{\mathbb{Q}}$, the set of $A$-rational points is finite, i.e. $X(A)$ is finite. In this Paper, the mention a conjecture (Conjecture 1.1), in which claims that $X$ is arithmetically hyperbolic over $\overline{\mathbb{Q}}$ if and only if for every abelian variety $A$ over $\overline{\mathbb{Q}}$, every morphism $A\rightarrow X$ over $\overline{\mathbb{Q}}$ is constant.
However, I am confused what this would mean in the case that $X$ is an abelian variety itself. There exist elliptic curves of rank one over $\overline{\mathbb{Q}}$, i.e. $X(\mathbb{Z})$ is a finite torsion group. However, the map $X\rightarrow X, x\mapsto 2x$ is not constant. Therefore $X$ should not have only finitely many integral points. Since this seems like an obvious counterexample, I must misunderstand something very basic about this conjecture. What is it?