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Direct CP asymmetries (${\cal A}_{CP}$) in the inclusive decays of${\overline B}\to X_s\gamma$ and ${\overline B}\to X_{s+d}\gamma$ of the orderof $1\%$ will be probed at the BELLE II experiment. In this work, three suchasymmetries are studied in the context of a three-Higgs-doublet model (3HDM),and it is shown that all three ${\cal A}_{CP}$ can be as large as the currentexperimental limits. Of particular interest is ${\cal A}_{CP}$ for ${\overlineB}\to X_{s+d}\gamma$, which is predicted to be effectively zero in the StandardModel (SM). A measurement of $2.5\%$ or more for this observable with the fullBELLE II data would give $5\sigma$ evidence for physics beyond the SM. Wedisplay parameter space in the 3HDM for which such a clear signal is possible.

CP asymmetries of ${\overline B}\to X_s/X_d\gamma$ in models with three Higgs doublets