Equivalence of the Paschke morphism p_X under minimal hypotheses

Determine whether the Paschke morphism p_X: K^X(X) → K^{an}(X) is an equivalence for arbitrary locally compact uniform bornological coarse spaces X, beyond the current hypothesis that X is homotopy equivalent to a countable, finite-dimensional, locally finite simplicial complex with the spherical path metric. Establish precise conditions under which p_X induces isomorphisms on all homotopy groups.

Background

The paper constructs a Paschke duality transformation p_X: K^X(X) → K^{an}(X) relating the local coarse K-homology K^X(X) to analytic locally finite K-homology K^{an}(X). Theorem zhpoergtrghrrzhrzrjz asserts that p_X is an equivalence when X is homotopy equivalent to a countable, finite-dimensional, locally finite simplicial complex equipped with the spherical path metric.

Classical Paschke duality (in the sense of Higson–Roe) is known to be an isomorphism for any locally compact metric space, but the present morphism has different domain and codomain. The authors explicitly note uncertainty about removing the finite-dimensional locally finite simplicial complex assumption for their morphism p_X, indicating a gap in understanding the general scope of Paschke duality in the coarse-local framework.