CMC existence conjecture for future timelike geodesically complete cosmological spacetimes

Let $(M,g)$ be a spacetime with compact Cauchy surfaces. A CMC Cauchy surface is a spacelike Cauchy surface with constant mean curvature. The strong energy condition is the condition

$$\emph{Ric}(X,X) \geq 0$$

for all timelike vectors $X$.

CMC existence conjecture. If $(M,g)$ is future timelike geodesically complete and satisfies the strong energy condition, then $(M,g)$ contains a CMC Cauchy surface.

A theorem establishes this conclusion under the stronger assumption of everywhere nonpositive timelike sectional curvature. The conjecture asks whether the strong energy condition alone suffices; its resolution is not specified in the supplied text.