  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.
