Future timelike completeness conjecture for CMC Cauchy surfaces

Let $(M,g)$ be a spacetime with compact Cauchy surfaces. 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. This conjecture is motivated by Bartnik's CMC existence theorem and the result that non-positive timelike sectional curvature gives the required compactness condition. The source notes partial support in a natural cosmological context and verifies the conjecture for spacetimes admitting a future-complete timelike conformal Killing vector field.