Large-scale geometric reconstruction conjecture for stably causal Lorentzian manifolds

Let $M$ be a stably causal Lorentzian manifold whose sectional curvatures are much less than $1$, and let $\gamma$ be a timelike curve from $s$ to $t$ in $M$ whose radius of curvature and length are large. In the causal site of Example~, consider the causal paths $A_1 \prec \cdots \prec A_m$ such that each region $A_i$ intersects $\gamma$. Large-scale geometric reconstruction conjecture. The proper length of $\gamma$ is well approximated by the least upper bound of the lengths of these causal paths, with an error which is small compared to the length of $\gamma$. This conjecture asserts that causal sites retain large-scale geometric information about curved spacetimes: when curvature and the curve's radius of curvature are small relative to the Planck scale, causal-path lengths recover the curve's proper length up to a comparatively small error. The source provides no resolution of the conjecture.