The stability of future rays under perturbation of vacuum initial data

Let $(M,g,K)$ be a compact, vacuum initial data set whose evolution is a non-static spacetime. Suppose there is a family of future rays intersecting the initial Cauchy surface in an open set. Future-ray stability conjecture. Any sufficiently small $C^{2,\alpha}$ perturbation of the initial data evolves into a spacetime with at least one future ray. This conjecture would support the expectation that small perturbations of Bartnik-type data retain a global future ray and therefore need not produce CMC slices. Its general validity is open.