Bonnet-Myers rigidity theorem for globally hyperbolic Lorentzian length spaces

We prove a synthetic Bonnet-Myers rigidity theorem for globally hyperbolic Lorentzian length spaces with global curvature bounded below by $K<0$ and an open distance realizer of length $L=\frac{\pi }{\sqrt{\mathrm{\mid }K\mathrm{\mid }}}$: It states that the space necessarily is a warped product with warping function $\cos :(-\frac{\pi }{2},\frac{\pi }{2})\to {\mathbb{R}}_{+}$. From this, one also sees that a globally hyperbolic spacetime with curvature bounded above by $K<0$ and an open distance realizer of length $L=\frac{\pi }{\sqrt{\mathrm{\mid }K\mathrm{\mid }}}$ is a warped product with warping function $\cos$.