We prove that a globally hyperbolic smooth spacetime endowed with a $\mathrm{C}^1$-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in some distributional sense, obeys the timelike measure-contraction property $\textsf{TMCP}_p$. This embeds the class of spaces of lowest regularity for which classical singularity theorems are known into the synthetic Lorentzian setting of Cavalletti-Mondino and then explored in more depth by Braun. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption.
If the metric is even $\mathrm{C}^{1,1}$, in fact the stronger timelike curvature-dimension condition $\textsf{TCD}_p$ holds. In this regularity, we also obtain uniqueness of chronological $\ell_p$-geodesics and $\ell_p$-optimal couplings.