Bartnik's stationary vacuum conjecture for mass minimizers

Let $(\Omega, g_0, k_0)$ be a compact initial data set with nonempty smooth boundary, satisfying the dominant energy condition, and let $(M, g, k)$ be a Bartnik mass minimizer for $(\Omega, g_0, k_0)$. Bartnik's stationary vacuum conjecture. The interior initial data $(\operatorname{Int} M, g, k)$ should embed in a vacuum spacetime admitting a timelike Killing vector field; equivalently, $(M, g, k)$ should embed in a vacuum and strongly stationary spacetime. The conjecture concerns the expected geometric characterization of Bartnik mass minimizers; it was originally formulated in three dimensions and can naturally be posed in general dimensions. The pp-wave counterexamples discussed by Huang and Lee disprove it for zero Bartnik mass in dimensions $n>4$.