  CP4 3HDM is a three-Higgs-doublet model based on the $CP$ symmetry of order 4
(CP4). Imposing CP4 leads to remarkable connections between the scalar and
Yukawa sectors and unavoidably generates tree-level flavor-changing neutral
couplings (FCNC). It remains unclear whether FCNC can be sufficiently
suppressed in the CP4 3HDM. In this paper, we systematically explore this
issue. We first develop an efficient scanning procedure which takes the quark
masses and mixing as input and expresses the FCNC matrices in terms of physical
quark observables and quark rotation parameters. This procedure allows us to
explore the FCNC effects for all the Yukawa sectors possible within the CP4
3HDM. We find that, out of the eight possible CP4 Yukawa sectors, only two
scenarios are compatible with the $K$, $B$, $B_s$ and, in particular, $D$-meson
oscillation constraints. The results of this work serve as clear guidelines for
future phenomenological scans of the model.
