\begin{document}$H_f\,( f = u_1, d, e)$\end{document}. The conditions for anomaly cancellation imply that the remaining Higgs doublet of the model (\begin{document}$H_{u_2}$\end{document}) must have the same hypercharge as \begin{document}$H_{u_1}$\end{document}, and thus, can only couple to up-type quarks, which opens the possibility to have FCNCs only in this sector. We study the Lagrangian of the model, and in particular, the Higgs potential, to identify the Higgs mass-eigenstates and their interactions; for the Yukawa matrices, we consider the four-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 the parameter space. Subsequently, these constraints are used to evaluate the prospects for detecting the FCNC decay mode \begin{document}$t \to ch$\end{document} at the future high-luminosity (HL) option for the LHC, which are compared with current limits from LHC-run2. Moreover, we evaluate the FCNC decay of the next heavier Higgs boson \begin{document}$H_2\to tc$\end{document}, which can typically reach \begin{document}$BR(H_2 \rightarrow tc) \approx {\cal{O}}(10^{-4}-10^{-5} )$\end{document}. The search for the signal at HL-LHC is also studied, and it is found that it may be detectable for specific regions of the model parameter space."> A private SUSY 4HDM with FCNC in the up-sector -
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