This paper presents a search for direct top squark pair production in events with missing transverse momentum plus either a pair of jets consistent with Standard Model Higgs boson decay into $b$-quarks or a same-flavour opposite-sign dilepton pair with an invariant mass consistent with a $Z$ boson. The analysis is performed using the proton$-$proton collision data at $\sqrt{s}=13$ TeV collected with the ATLAS detector during the LHC Run-2, corresponding to an integrated luminosity of 139 fb$^{-1}$. No excess is observed in the data above the Standard Model predictions. The results are interpreted in simplified models featuring direct production of pairs of either the lighter top squark ($\tilde{t}_1$) or the heavier top squark ($\tilde{t}_2$), excluding at 95% confidence level $\tilde{t}_1$ and $\tilde{t}_2$ masses up to about 1220 and 875 GeV, respectively.

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