Uniform Lorentz-distance equi-Lipschitzness for all Cauchy temporal functions

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 $\tau\in C^\infty(M)$ and its level sets

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

consider the families of Lorentz distances $\{\ell(\cdot,\Sigma_t)\}_{t>0}$ and $\{\ell(\Sigma_{-t},\cdot)\}_{t>0}$. The universal equi-Lipschitz conjecture. For every Cauchy temporal function $\tau$, both of the families $\{\ell(\cdot,\Sigma_t)\}_{t>0}$ or $\{\ell(\Sigma_{-t},\cdot)\}_{t>0}$ are locally equi-Lipschitz. This stronger conjecture is proposed because it would imply the preceding existence conjecture. The supplied text gives no resolution, so its status remains open.