A search for new heavy scalars with flavour-violating decays in final states with multiple leptons and $b$-tagged jets is presented. The results are interpreted in terms of a general two-Higgs-doublet-model involving an additional scalar with couplings to the top-quark and the three up-type quarks ($\rho_{tt}$, $\rho_{tc}$, and $\rho_{tu}$). The targeted signals lead to final states with either a same-sign top-quark pair, three top-quarks, or four top-quarks. The search is based on a dataset of proton-proton collisions at $\sqrt{s}=13$ TeV recorded with the ATLAS detector during Run 2 of the Large Hadron Collider, corresponding to an integrated luminosity of 139 fb$^{-1}$. Events are categorised depending on the multiplicity of light charged leptons (electrons or muons), total lepton charge, and a deep-neural-network-based categorisation to enhance the purity of each of the signals. A mild excess is observed over the Standard Model expectation corresponding to a local significance of 2.81 standard deviations for a signal with $m_{H}=1000$ GeV and $\rho_{tt}=0.32$, $\rho_{tc}=0.05$, and $\rho_{tu}=0.85$. Exclusion limits at 95% confidence are set on the mass and couplings of the heavy Higgs boson. Masses of an additional scalar boson $m_{H}$ between 200-630 (200-840) GeV with couplings $\rho_{tt}=0.4$, $\rho_{tc}=0.2$, and $\rho_{tu}=0.2$ are observed (expected) to be excluded at 95% confidence level. Additional interpretations are provided in models of $R$-parity violating supersymmetry, motivated by the recent flavour and $(g-2)_\mu$ anomalies.

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