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 sense of Cavalletti and Mondino. We prove existence geodesics probability measures on M which satisfy entropic semiconvexity inequality defining whose densities with respect to m are additionally uniformly L∞ time. This holds apart from any nonbranching assumption. also discuss similar results under measure-contraction property.