The constant signed mean-curvature conjecture

Let $\mathcal{V}$ be a cosmological spacetime. A slice has constant signed mean curvature if its mean curvature is constant with a fixed sign convention, while a CMC slice is a slice of constant mean curvature. Constant signed mean-curvature conjecture. The spacetime $\mathcal{V}$ has a CMC slice if and only if it has a slice of constant signed mean curvature. The conjecture strengthens the obstruction results for Bartnik-type spacetimes, which rule out slices of constant signed mean curvature as well as CMC slices. Its general validity is open.