  We study the simplest generic extension of the Standard Model which allows
for conventional electroweak baryogenesis, through the addition of dimension
six operators in the Higgs sector. At least one such operator is required to be
CP-odd, and we study the constraints on such a minimal setup, and related
scenarios with minimal flavor violation, from the null results of searches for
electric dipole moments (EDMs), utilizing the full set of two-loop
contributions to the EDMs. The results indicate that the current bounds are
stringent, particularly that of the recently updated neutron EDM, but fall
short of ruling out these scenarios. The next generation of EDM experiments
should be sufficiently sensitive to provide a conclusive test.
