Busemann's manifold conjecture for Busemann $G$-spaces

Let $(X,d)$ be a Busemann $G$-space, meaning that it is a metric space satisfying Menger convexity, finite compactness, local extendibility, and uniqueness of extension. Suppose that $X$ has dimension $n\in\mathbb{N}$. Busemann's conjecture. $X$ is a topological $n$-manifold. This conjecture concerns the topological structure forced by the geodesic and local extension properties of Busemann $G$-spaces. It remains open in dimensions $n\geq 5$.