  We address the following question. Take the Constrained Minimal
Supersymmetric Standard Model (CMSSM) with the two CP violating SUSY phases
different from zero, and neglect the bound coming from the electric dipole
moment of the neutron (EDM): is it possible to fully account for CP violation
in the kaon and B systems using only the SUSY contributions with vanishing CKM
phase? We show that the ${BR}(B\to X_{s} \gamma)$ constraint, though CP
conserving, forces a negative answer to the above question. This implies that,
even in the regions of the CMSSM where a cancellation of different
contributions to the EDM allows for large SUSY phases, it is not possible to
exploit the SUSY phases to fully account for observable CP violation. Hence to
have sizeable SUSY contributions to CP violation, one needs new flavor
structures in the sfermion mass matrices beyond the usual CKM matrix.
