Consider a metric space $(X,d)$ with a distinguished selection of geodesics, i.e. a [geodesic bicombing][1] $\sigma:X\times X\times [0,1]\rightarrow X$.  We call a geodesic bicombing *conical* if it satisfies the following "convexity-type condition"
$$
d(\sigma_{xy}(t), \sigma_{x^\prime y^\prime}(t))\leq (1-t)d(x,x^\prime)+t d(y,y^\prime)
$$
for all $x,x',y,y'\in X$ and $t\in [0,1]$.  

---
If $(X,d)$ is a $\delta$-hyperbolic intrinsic metric space, for some $\delta>0$, then does $(X,d)$ admit a conical geodesic bicombing?


  [1]: https://en.wikipedia.org/wiki/Geodesic_bicombing