  1T-TaS$_2$ is a charge-density-wave (CDW) compound with a Mott-insulating
ground state. The metallic state obtained by doping, substitution or pulsed
charge injection is characterized by an emergent CDW domain wall network, while
single domain walls can be found in the pristine Mott state. Here we study
whether and how the single walls become metallic. Tunneling spectroscopy
reveals partial suppression of the Mott gap and the presence of in-gap states
strongly localized at the domain-wall sites. Using the real-space dynamical
mean field theory description of the strongly correlated quantum-paramagnet
ground state we show that the local gap suppression follows from the increased
hopping along the connected zig-zag chain of lattice sites forming the domain
wall, and that full metallisation is preempted by the splitting of the
quasiparticle band into bonding and antibonding sub-bands due to the structural
dimerization of the wall, explaining the presence of the in-gap states and the
low density of states at the Fermi level.
