The open contact submanifold conjecture for spaces of light rays

Let $(X,g)$ be a causally simple spacetime that can be conformally embedded as an open subset into some globally hyperbolic spacetime $(\widetilde X,\widetilde g)$. Let the space of light rays mean the space of future-directed, unparameterized light rays. The open contact submanifold conjecture. The space of light rays in $(X,g)$ is an open contact submanifold of the space of light rays in $(\widetilde X,\widetilde g)$. This conjecture concerns the compatibility of the contact geometry of light-ray spaces under conformal embeddings and is proposed as an ingredient toward the causally simple Legendrian Low conjecture; it remains open.