  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 $L=\frac{\pi}{\sqrt{|K|}}$. In the course of
the proof, we show that the space necessarily is a warped product with warping
function $\cos:(-\frac{\pi}{2},\frac{\pi}{2})\to\mathbb{R}_+$.
