Initial data sets with dominant energy condition admitting no smooth DEC spacetime extension

There are two versions of the dominant energy condition (=DEC): The original one for Lorentzian manifolds and an associated one for initial data sets. If a Lorentzian manifold satisfies DEC, then so does the induced initial set on any embedded spacelike hypersurface. In this article, we discuss the question of a potential converse of this: Is every DEC initial data set the induced one on a spacelike hypersurface within a suitably chosen DEC Lorentzian manifold? We provide an example showing that in general the answer is no if we require all structures to be smooth.