Dilts–Holst conjecture on CMC Cauchy surfaces

Let $(M,g)$ be a spacetime with compact Cauchy surfaces satisfying the strong energy condition. Dilts–Holst conjecture. If $(M,g)$ has a Cauchy surface of constant signed mean curvature, then $(M,g)$ contains a CMC Cauchy surface. The conjecture is motivated by Bartnik's nonexistence example, which rules out Cauchy surfaces whose mean curvature is everywhere strictly positive or everywhere strictly negative, together with related considerations. The source presents a subsequent result as partial support for this conjecture.