Horofunction compactifications and local Gromov model domains

We explore the horofunction compactification of complete hyperbolic domains in complex Euclidean space equipped with the Kobayashi distance. We provide a sufficient condition under which, given a domain $\Omega$ as above, the identity map from $\Omega$ to itself extends to an embedding of $\overline{\Omega}$ into the horofunction compactification of $(\Omega,k_\Omega)$, with $k_\Omega$ denoting the Kobayashi distance on $\Omega$. Notably, this condition admits unbounded domains that are not Gromov hyperbolic relative to the Kobayashi distance. We also provide a large class of planar hyperbolic domains satisfying the above condition.