Berestovski\s conjecture on locally compact similarity-homogeneous metric spaces

Let $(X,\rho)$ be a locally compact, similarity-homogeneous, non-homogeneous metric space with inner metric. Let $F$ be an arbitrary level set of the function measuring the radius of completeness on $X$. Write $\operatorname{Sim}(X)$ for the topological group of similarities of $X$, and let $\operatorname{Isom}(X)\subset\operatorname{Sim}(X)$ be its subgroup of isometries.

Berestovski\s conjecture. The space $X$ is homeomorphic to the topological product

$$F\times\mathbb R_+,$$

and

$$\operatorname{Sim}(X)\cong\operatorname{Isom}(X)\times\mathbb R_+.$$

The conjecture describes the product structure of such metric spaces and the corresponding decomposition of their similarity groups. The paper constructs an example showing that the local compactness hypothesis is essential; no resolution beyond that limitation is stated here.