  We present a SUSY model with four Higgs doublets of the ``private type",
where each fermion type (up, down, and charged leptons) obtain their masses
from a different Higgs doublet H_f ( f = u_1, d, e). The conditions for anomaly
cancellation, imply that the remaining Higgs doublet of the model (H_{u_2}),
must have the same hypercharge as H_{u_1}, and thus can only couple to up-type
quarks, which opens the possibility to have FCNC's only in this sector. We
study the Lagrangian of the model, and in particular the Higgs potential, in
order to identify the Higgs mass-eigenstates and their interactions; for the
Yukawa matrices we consider the 4-texture case. We obtain constraints on the
model parameters by using LHC measurements on the properties of the 125 GeV
Higgs boson ($h$), and identify viable regions of parameter space. These
constraints are then used to evaluate the prospects to detect the FCNC decay
mode t -> ch at the future high-luminosity (HL) option for the LHC, which are
compared with current limits from LHC-run2. Then, we also evaluate the FCNC
decay of the next heavier Higgs boson H_2 -> tc, which can reach typically
BR(H_2 -> tc) = O(10^{-4}-10^{-5} ); the search for the signal at HL-LHC is
also studied, finding that it could be detectable for some specific regions of
the model parameter space.
