Legendrian-linking conjecture for causality in (3+1)-dimensional spacetimes

Let $(M,g)$ be a globally hyperbolic $(3+1)$-dimensional spacetime with a Cauchy surface diffeomorphic to a subset of $\mathbb{R}^3$, and let $N$ be its manifold of light rays. For spacetime points, let their skies be the corresponding Legendrian submanifolds of $N$. Legendrian-linking conjecture. Two spacetime points are causally related in $M$ if and only if their skies either intersect or are Legendrian linked in $N$. In contrast with smooth linking, Legendrian linking distinguishes past from future in the Minkowski $(2+1)$-dimensional example discussed in the paper, suggesting that it is the appropriate invariant for the higher-dimensional causality problem.