CPconserving constraints on supersymmetricCPviolation in the MSSM

We address the following question. Take the constrained minimal supersymmetric standard model (CMSSM) with the two $\mathrm{CP}$ violating supersymmetry (SUSY) phases different from zero, and neglect the bound coming from the electric dipole moment (EDM) of the neutron: is it possible to fully account for $\mathrm{CP}$ violation in the kaon and B systems using only the SUSY contributions with a vanishing CKM phase? We show that the $\mathrm{BR}\left(\overrightarrow{B}{X}_s\gamma \right)$ constraint, though $\mathrm{CP}$ 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 $\mathrm{CP}$ violation. Hence to have sizable SUSY contributions to $\mathrm{CP}$ violation, one needs new flavor structures in the sfermion mass matrices beyond the usual CKM matrix.