The open-set conjecture for initial data without CMC slices

Let $(M,g,K)$ be an initial data set, equipped with the $C^{2,\alpha}$ norm, and let its associated spacetime be the evolution determined by the data. Open-set conjecture. There is an open set of initial data in the $C^{2,\alpha}$ norm such that the associated spacetimes do not have CMC slices. The claim concerns genericity in the topology of initial data rather than a global spacetime topology. The paper explains that existence of CMC slices is open, but conjectures that it is not dense; the status of this non-density assertion remains open.