  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 data sample 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 139f b$^{-1}$.
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. Masses of an
additional scalar boson $m_{H}$ between $200-630$ GeV with couplings
$\rho_{tt}=0.4$, $\rho_{tc}=0.2$, and $\rho_{tu}=0.2$ are 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.
