Coarse equivalence from homeomorphic Higson coronas under positive asymptotic dimension

Determine whether, for proper metric spaces X and Y with positive asymptotic dimension, the homeomorphism of their Higson coronas νX and νY implies that X and Y are coarsely equivalent.

Background

The paper establishes Banach–Stone-like results for lattices of slowly oscillating functions on chain-connected proper metric spaces, showing that lattice isomorphisms induce coarse homeomorphisms and homeomorphisms of Higson compactifications. It is known that coarsely equivalent proper metric spaces have homeomorphic Higson coronas.

However, the converse implication from homeomorphic Higson coronas to coarse equivalence fails in general. Motivated by the fact that any unbounded chain-connected proper metric space has positive asymptotic dimension, the authors ask whether adding the assumption of positive asymptotic dimension bridges the gap, i.e., whether homeomorphic Higson coronas force coarse equivalence under this condition.