Good geodesics satisfying the timelike curvature-dimension condition

Let (M,d,m,≪,≤,τ) be a causally closed, K-globally hyperbolic, regular measured Lorentzian geodesic space satisfying the weak timelike curvature-dimension condition wTCDpe(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 wTCDpe(K,N) and whose densities with respect to m are additionally uniformly L∞ in time. This holds apart from any nonbranching assumption. We also discuss similar results under the timelike measure-contraction property.