Bartnik's cosmological splitting conjecture in terms of Lorentz-distance families

Let $(M,g)$ be a globally hyperbolic, timelike geodesically complete spacetime with compact Cauchy surfaces satisfying the strong energy condition. For a Cauchy temporal function $\tau a$ and its level sets

$$\Sigma_t:=\{\tau=t\},$$

consider the families of Lorentz distances $\{\ell(\cdot,\Sigma_t)\}_{t>0}$ and $\{\ell(\Sigma_{-t},\cdot)\}_{t>0}$. Bartnik's cosmological splitting conjecture. There exists a Cauchy temporal function $\tau\in C^\infty(M)$ such that one of these two families is locally equi-Lipschitz. This formulation is stated as equivalent to Bartnik's cosmological splitting conjecture; the paper's counterexamples show that compactness of Cauchy surfaces and the strong energy condition cannot independently be omitted, so the conjecture is refuted by the counterexamples discussed in the paper.