Orbit transversality for abelian varieties

The dynamical Mordell–Lang problem asks how one forward orbit intersects a subvariety. We consider a complementary question over number fields, allowing passage to a suitable finite extension: can a proper Zariski-closed subset meet every rational grand orbit?
Assuming a Zariski-dense forward orbit, Pasten and Silverman answered this question negatively for linear automorphisms of projective space and for self-morphisms of geometrically simple abelian varieties. We will see that the answer is no for every dominant self-morphism of an abelian variety, without the dense-orbit assumption. If time permits, we will also discuss the case of projective space.