The causal Legendrian-linking conjecture outside the $Y^x_{\ell}$ case

Let $X$ be a globally hyperbolic spacetime with Cauchy surface $M$. A $Y^x_{\ell}$ Riemann metric on $M$ is a Riemannian metric for which there exist a point $x$ and a positive number $\ell$ such that all unit-speed geodesics from $x$ return to $x$ in time $\ell$. For events $x,y\in X$, let $\mathfrak S_x$ and $\mathfrak S_y$ denote their Legendrian spheres of light rays.

Causal Legendrian-linking conjecture. Assume that one cannot put a $Y^x_{\ell}$ Riemann metric on the Cauchy surface $M$. Then two events $x,y\in X$ are causally related if and only if the Legendrian link $(\mathfrak S_x,\mathfrak S_y)$ is non-trivial.

This conjecture proposes an equivalence between causal relatedness and non-trivial Legendrian linking for globally hyperbolic spacetimes whose Cauchy surfaces do not admit a $Y^x_{\ell}$ Riemann metric. The paper notes that it is not currently known whether every compact simply connected manifold with the integral cohomology ring of a CROSS admits such a metric; the conjecture is intended to cover the remaining higher-dimensional cases beyond the known sphere-quotient situation.