A search is made for a vector-like $T$ quark decaying into a Higgs boson and a top quark in 13 TeV proton-proton collisions using the ATLAS detector at the Large Hadron Collider with a data sample corresponding to an integrated luminosity of 139 fb$^{-1}$. The Higgs-boson and top-quark candidates are identified in the all-hadronic decay mode, where $H\rightarrow b\bar{b}$ and $t\rightarrow b W \rightarrow b q \bar{q}^\prime$ are reconstructed as large-radius jets. The candidate Higgs boson, top quark, and associated B-hadrons are identified using tagging algorithms. No significant excess is observed above the background, so limits are set on the production cross-section of a singlet $T$ quark at 95% confidence level, depending on the mass, $m_T$, and coupling, $\kappa_T$, of the vector-like $T$ quark to Standard Model particles. In the considered mass range between 1.0 and 2.3 TeV, the upper limit on the allowed coupling values increases with $m_T$ from a minimum value of 0.35 for $1.07 < m_T < 1.4$ TeV to 1.6 for $m_T = 2.3$ TeV.

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