  Let $(M,\mathsf{d},\mathfrak{m},\ll,\leq,\tau)$ be a causally closed,
$\mathscr{K}$-globally hyperbolic, regular measured Lorentzian geodesic space
satisfying the weak timelike curvature-dimension condition
$\smash{\mathrm{wTCD}_p^e(K,N)}$ in the sense of Cavalletti and Mondino. We
prove the existence of geodesics of probability measures on $M$ which satisfy
the entropic semiconvexity inequality defining $\smash{\mathrm{wTCD}_p^e(K,N)}$
and whose densities with respect to $\mathfrak{m}$ are additionally uniformly
$L^\infty$ in time. This holds apart from any nonbranching assumption. We also
discuss similar results under the timelike measure-contraction property.
