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Can future/past d-distinction be weakened to weak d-distinction in the distance-based characterization of global hyperbolicity?

Determine whether Theorem \ref{cakop}, which characterizes global hyperbolicity of a smooth spacetime (M,g) using properties of the Lorentzian distance d—including condition (iii) "future or past d-distinction" (i.e., either d_p = d_q implies p = q for all p,q in M, or d^p = d^q implies p = q for all p,q in M)—remains valid when condition (iii) is replaced by the weaker "weak d-distinction" requirement (namely, for all p,q in M, both d_p = d_q and d^p = d^q together imply p = q).

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Background

Theorem \ref{cakop} provides a formulation of global hyperbolicity purely in terms of the Lorentzian distance d, requiring (i) finiteness of d, (ii) existence of a topology T making d continuous with chronological diamonds relatively compact, and (iii) either future or past d-distinction. In prior work on Lorentzian metric spaces, the distinguishing condition used is the weaker "weak d-distinction" (both forward and backward distance profiles matching implies equality of points).

The authors note a gap between their current use of future/past d-distinction and the weaker condition adopted in earlier abstract frameworks. The open question asks whether the stronger assumption in (iii) can be relaxed to weak d-distinction without losing the equivalence in Theorem \ref{cakop}.

References

Taking into account that d satisfies the reverse triangle inequality for chronologically related events and that every event is included in some chronological diamond, we recognize that these properties are precisely those that define a Lorentzian metric space (without chronological boundary, M=I(M)) in the sense of our previous joint work with Suhr [] [Thm.\ 2.14], up to the fact that in a Lorentzian metric space (iii) is replaced by the slightly weaker `weak d-distinction': \forall p,q\in M, d_p=d_q and dp=dq \Rightarrow p=q (unfortunately, we do not know if (iii) can be weakened to such property).

— Global hyperbolicity and manifold topology from the Lorentzian distance (2503.04382 - Bykov et al., 6 Mar 2025) in Introduction, after Theorem \ref{cakop}