Are the auxiliary hypotheses in Theorem \ref{cjpr} necessary?

Ascertain whether the additional hypotheses in Theorem \ref{cjpr}—namely (a) d-reflectivity, (b) T-continuity of all functions d_p and d^p together with p ∈ \overline{I^{\pm}(p)} in the topology T, or (c) strong causality—can be removed while still ensuring that the topology T coincides with both the Alexandrov and manifold topologies and that (M,g) is globally hyperbolic.

Background

Theorem \ref{cjpr} gives a topology-independent characterization of global hyperbolicity based solely on properties of the Lorentzian distance and chronological diamonds. To conclude that the chosen topology T coincides with the manifold/Alexandrov topology, the theorem assumes one of three auxiliary conditions: d-reflectivity, continuity of distance profiles plus closure of chronological futures/pasts, or strong causality.

The authors indicate that these auxiliary assumptions are used to prevent T from being "too fine," but it remains unknown whether they are strictly necessary. Removing them would strengthen the topology-free characterization of global hyperbolicity.