With the following **additional** assumptions: $K$ is compact, *connected and locally convex*, then it is proved in https://arxiv.org/pdf/1304.4147.pdf that $K$ is convex (Theorem 1.1). In this case, $co(K)=K$ is compact.

Carlos Ramos-Cuevas: Convexity is a local property in CAT(κ) spaces, [arXiv:1304.4147](https://arxiv.org/abs/1304.4147)