Is the answer to the following problem, or some close variant thereof, known? Briefly:

• Given two initial data sets $I_1=(M,g_1,k_1)$ and $I_2=(M,g_2,k_2)$, is there a time-oriented spacetime satisfying the dominant energy condition which has both $I_1$ and $I_2$ as Cauchy surfaces? i.e. can two initial data sets be filled in by the dominant energy condition?

In more detail:

Let $M$ be a smooth manifold, $g_1$ and $g_2$ two Riemannian metrics and $k_1$ and $k_2$ two symmetric 2-tensors.

Does there necessarily exist a time-oriented Lorentzian manifold $(N,g')$ satisfying the dominant energy condition into which there are hypersurface embeddings $f_1$ and $f_2$ of $M$ such that:

• $f_1^\ast g'=g_1$ and $f_2^\ast g'=g_2$
• $k_1$ and $k_2$ are the second fundamental forms (relative to the time-oriented unit normal) of $f_1$ and $f_2$
• every maximally extended timelike geodesic of $(N,g')$ intersects both $f_1(M)$ and $f_2(M)$ exactly once?

This seems to be somewhat analogous to a question studied in the Riemannian setting, see e.g. the article of Xue Hu and Yuguang Shi here https://www.emis.de/journals/SIGMA/Gromov.html