Suppose that $(X,d)$ is a locally compact connected homogeneous metric space, where by homogeneous I mean that for any $x_0,x_1 \in X$ there exists an isometry $f:X\rightarrow X$ such that $f(x_0)=x_1$. Does there necessarily exist a connected manifold $M$ and a continuum (compact connected metric space) $K$ such that $X$ and $M\times K$ (with the max metric) are bi-uniformly equivalent, i.e. there exists a bijection $g:M\times K \rightarrow X$ such that $g$ and $g^{-1}$ are both uniformly continuous. If so can $M$ and $K$ be taken to be homogeneous themselves?

When I looked for an answer to this question the only thing I could find was this paper of Berestovskii, which, if I'm understanding it correctly, gives a positive answer with the additional assumption that the space has a geodesic metric.

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