The Legendrian Low conjecture for causally simple spacetimes

Let $(X^{m+1},g)$, $m>1$, be a causally simple spacetime that does not admit a metric $g'$ such that $(X,g')$ is refocussing and causally simple. A Legendrian link is a link formed by a pair of skies in the space of light rays of $(X,g)$. The Legendrian Low conjecture. The Legendrian links formed by a pair of skies of two causally unrelated events in $(X,g)$ and by a pair of skies of two causally related events in $(X,g)$ that do not belong to the same null geodesic are never Legendrian isotopic. This conjecture extends the relation between causality and Legendrian linking beyond the globally hyperbolic setting; the preceding corollary establishes a restricted version, but the full assertion remains open.