Recovering the metric of a $CAT(0)$-space by a diagonal tube

Let $(x,d)$ be a locally compact geodesically complete $CAT(0)$-space of topological dimension $>1$. It is proved that if each geodesic segment in $X$ admits a unique continuation to a complete geodesic, then the metric $d$ is recovered by the diagonal tube $V\subset X\times X$ corresponding to an arbitrary $r>0$. This partly generalizes V. N. Berestovskii's results on A. D. Aleksandrov spaces of negative curvature.