Compactification equivalence for a Gromov hyperbolic strip-minus-closed-set planar domain

Determine whether, for the planar domain D = {z in C : |Im z| < 1} \ C where C is a closed subset of the strip satisfying i(C) = C, h(C) = C for i(z) = z̄ and h(z) = z + 1, C ∩ R = ∅, and D is connected, the horofunction compactification of (D, k_D) is equivalent to its Gromov compactification.

Background

Section 5 constructs a planar domain D obtained by removing a closed, translation- and reflection-invariant set C from the horizontal strip of height 2. The authors show that (D, k_D) is Gromov hyperbolic and does not have the approaching geodesics property by constructing a hyperbolic isometry with minimal displacement strictly larger than its divergence rate.

They end by explicitly asking whether, for this specific domain, the horofunction and Gromov compactifications coincide.