CP4 3HDM is a three-Higgs-doublet model based on a symmetry of order 4 (CP4). It is the minimal model incorporating CP4 without leading to accidental symmetries or running into immediate conflict with experiment. Imposing CP4 on the lagrangian induces remarkably tight connections between the scalar and Yukawa sectors, including the unavoidable tree-level flavor-changing neutral couplings (FCNC). Here, we explore whether it is at all possible in the CP4 3HDM to suppressed FCNC to a level compatible with the neutral meson oscillation constraints. We express the FCNC matrices in terms of physical quark observables and quark rotation parameters, and scan the Yukawa parameter space using the quark masses and mixing parameters as input. With this procedure, we find that only two out of the eight possible CP4 Yukawa sectors are compatible with the , , and, in particular, -meson oscillation constraints. The results clearly indicate a way how to construct phenomenologically viable benchmark CP4 3HDMs.