Approaching geodesics and compactification equivalence for smooth pseudoconvex finite type domains in C^2

Determine whether, for any smooth pseudoconvex finite type domain D in C^2 equipped with the Kobayashi distance k_D, the metric space (D, k_D) has the approaching geodesics property, and ascertain whether the horofunction compactification of (D, k_D) is equivalent to its Gromov compactification.

Background

Section 3 surveys when the approaching geodesics property holds for domains in complex Euclidean space with the Kobayashi distance, and how this property relates to the equivalence of the horofunction and Gromov compactifications. For bounded strongly pseudoconvex domains with C^2 boundary, approaching geodesics are known to hold via the squeezing function, which in turn implies equivalence of compactifications.

For pseudoconvex finite type domains in C^2, the Gromov compactification is known to match the Euclidean compactification, but it is explicitly asked whether approaching geodesics hold in this class and whether the horofunction compactification coincides with the Gromov compactification.