The actual reconstruction input

If the completed event space is all of $V$, cone selection already gives a flat Lorentzian realization of its interior-cone chronology. A sparse sample such as Example ex:sparseLorentz is not thereby a manifold. A curved realization requires additional compatible local data.

Strong causality, not merely past and future distinguishability, is the condition relevant to equality of manifold and Alexandrov topologies ParrikarSurya. The path topology in HKM is not simply another name for the Alexandrov topology.

The Alexandrov and Zeeman rigidity results likewise concern transformations of an already specified causal geometry AlexandrovOvchinnikova,Zeeman. They are not used to select a cone from scheduler consistency.