Bartnik Splitting Conjecture for globally hyperbolic spacetimes

Let $M$ be a globally hyperbolic spacetime with compact Cauchy surfaces satisfying the strong energy condition. Bartnik Splitting Conjecture. If $M$ is timelike geodesically complete, then $M$ splits: $(M,g)$ is isometric to

$$({\mathbb R} \times S,-dt^2+h),$$

where $S$ is a smooth spacelike Cauchy hypersurface with induced metric $h$. The conjecture concerns rigidity in the Lorentzian analogue of the Cheeger–Gromoll splitting theorem and would imply that timelike geodesic completeness prevents singular behavior in this setting. Nevertheless, the conjecture remains open.