Decomposable abelian $G$-curves and special subvarieties

We consider families of abelian Galois coverings of the line. When the Jacobian of the general element is totally decomposable, i.e., is isogenous to a product of elliptic curves, we prove that they yield special subvarieties of ${\text{A}}_g$ if and only if a numerical condition holds, which in the general case is only known to be sufficient.