The conformal embedding conjecture for causally simple spacetimes

Let $(X,g)$ be a causally simple spacetime. A conformal embedding is an embedding preserving the conformal class of the metric. The conformal embedding conjecture. The spacetime $(X,g)$ can be conformally embedded as an open subset into some globally hyperbolic spacetime $(\widetilde X,\widetilde g)$. The source notes that all known examples have such an embedding and conjectures that this holds in general; the assertion remains open.