  We present a heuristic argument based on Honda-Tate theory against many
conjectures in `unlikely intersections' over the algebraic closure of a finite
field; notably, we conjecture that every abelian variety of dimension 4 is
isogenous to a Jacobian. Using methods of additive combinatorics, we are able
to give a negative answer to a related question of Chai and Oort where the
ambient Shimura Variety is a power of the modular curve.
