Approaching geodesics and compactification equivalence for Gromov hyperbolic bounded convex domains

Determine whether, for any Gromov hyperbolic bounded convex domain D in C^d 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

For certain convex settings, such as bounded convex finite type domains, the authors establish a weaker approaching complex geodesics property that suffices to prove equivalence of the horofunction and Gromov compactifications. If the bounded convex domain has smooth boundary, finite type follows and equivalence is known via this weaker property.

However, for general Gromov hyperbolic bounded convex domains, it remains explicitly asked whether approaching geodesics hold and whether the horofunction and Gromov compactifications coincide.