Bombelli's conjecture on Poisson reconstruction of distinguishing spacetimes

Let $M$ and $N$ be distinguishing finite-volume spacetimes. For each finite poset $C$, consider independent samples from Poisson point processes on $M$ and $N$, with the causal orders induced by the respective spacetimes.

Bombelli's conjecture. If, for every finite poset $C$, the probabilities that the samples from $M$ and $N$ are order-isomorphic to $C$ coincide, then $M$ and $N$ are isometric.

This is a rigid version of the Hauptvermutung in causal set theory, where the statistical data of Poisson samples is intended to reconstruct the underlying spacetime. A rigorous mathematical formulation of the broader Hauptvermutung remains missing, and the conjecture is presented as a first step toward such a reconstruction theory.