Completely (iso-)split scale-invariant Coulomb branch geometries are isotrivial

We show that scale-invariant special Kahler geometries whose generic r-complex-dimensional abelian variety fiber is isomorphic (completely split) or isogenous (completely iso-split) as a complex torus to the product of r one-dimensional complex tori have constant $\tau^{ij}$ modulus on the Coulomb branch, i.e., are isotrivial. These simple results are useful in organizing the classification of scale-invariant special Kahler geometries, which, in turn, is relevant to the classification of possible 4-dimensional N=2 supersymmetric superconformal field theories.