The ray-and-line criterion for CMC Cauchy surfaces

Let $\mathcal{V}$ be a globally hyperbolic, cosmological spacetime. A future ray or past ray is a complete timelike path in the corresponding time direction, and a line is a complete timelike geodesic extending in both directions. Ray-and-line criterion. The spacetime $\mathcal{V}$ has no CMC Cauchy surfaces if and only if it contains a future ray and a past ray, but no line. This conjecture is motivated by Bartnik's examples and by the splitting result that a spacetime containing a line is a metric product and therefore has maximal slices. The difficult direction is to establish CMC slices from the relevant timelike incompleteness condition; the general statement remains open.