  Motivated by the results of recent transport and optical conductivity
studies, we propose a semi-infinite two-dimensional lattice model for
interacting massive Dirac electrons in the pressurized organic conductor
$\alpha$-(BEDT-TTF)$_2$I$_3$, and address the problem of domain wall
conductivity in a charge-ordered insulating phase under realistic experimental
conditions. Using the extended Hubbard model at a mean field level, we present
results of extensive numerical studies around the critical region of the model,
reporting on the resistivity and optical conductivity calculated by means of
the Nakano-Kubo formula. We find that the activation gap extracted from the
resistivity data can be much smaller than the optical gap in the critical
region, which is induced by metallic conduction along an one-dimensional domain
wall emerging at the border of two charge-ordered ferroelectric regions with
opposite polarizations. The data are consistent with the observed transport gap
in real $\alpha$-(BEDT-TTF)$_2$I$_3$ samples that is reduced remarkably faster
than the optical gap upon suppressing charge order with pressure. Our optical
conductivity also reveals an additional shoulder-like structure at low energy
inside the gap, which is argued to be directly relevant to the metallic bound
states residing on the domain wall.
