The equivalence between timelike Ricci curvature and the timelike Brunn Minkowski inequality on synthetic Lorentzian spaces

We introduce the strong $q$-timelike Brunn-Minkowski condition ${\mathsf{s}\mathsf{T}\mathsf{B}\mathsf{M}}_q(K,N)$ on synthetic Lorentzian spaces, for $0<q<1$. We show that, in the timelike $q$-essentially non-branching setting, the $q$-timelike curvature dimension condition ${\mathsf{T}\mathsf{C}\mathsf{D}}_q(K,N)$ is equivalent to ${\mathsf{T}\mathsf{B}\mathsf{M}}_q(K,N^{+})$, and that the entropic $q$-timelike curvature dimension condition ${\mathsf{T}\mathsf{C}\mathsf{D}}_q^e(K,N)$ is equivalent to the reduced $\mathsf{s}\mathsf{T}\mathsf{B}\mathsf{M}$ condition, ${\mathsf{s}\mathsf{T}\mathsf{B}\mathsf{M}}_q^{\ast }(K,N)$. This extends, to a non-smooth setting, our earlier work in proving the equivalence between Ricci curvature and the Brunn-Minkowski inequality on $C^2$ spacetimes.