For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$.
Definition: A metric space $(X,d)$ is called
$\bullet$ medial if for any points $a,b\in X$ there exist unique points $x,y\in X$ such that $\mathbf Maxb$ and $\mathbf Maby$;
$\bullet$ symmetric if $\forall a,b,c,\alpha,\beta\in X\quad(\mathbf Mac\alpha\wedge \mathbf Mbc\beta)\Rightarrow d(a,b)=d(\alpha,\beta)$;
$\bullet$ a line space if $(X,d)$ is nonempty, medial and every bounded sequence in $X$ has a convergent subsequence.
It can be shown that for any distinct points $a,b\in X$ of a medial complete metric space $(X,d)$ there exists a unique isometric embedding $f:\mathbb R\to X$ such that $f(0)=a$ and $f(d(a,b))=b$. This fact implies that line metric spaces are Busemann G-spaces (but not vice versa).
Example: Every strictly convex finite-dimensional Banach space is a symmetric line space (and is homeomorphic to a Euclidean space $\mathbb R^n$).
The famous Busemann Conjecture claims that every Busemann G-space is a topological manifold. This conjecture is known to be true for Busemann G-spaces of topological dimension $\le 4$.
On the other hand, for symmetric line spaces the Busemann Conjecture seems to be true for all dimensions and in the following stronger form:
Conjecture: Every symmetric line space is homeomorphic to a Euclidean space $\mathbb R^n$.
I strongly believe that this Conjecture is true and can be derived from the known theory of homogeneous spaces of Lie groups and the Cartan classification of symmetric Riemannian manifolds. Nonetheless, I cannot find a precise reference in the literature.
So, my question:
Question. Is the Conjecture indeed true? And if yes, what is a proper reference?
Remark: The Conjecture is true if the answer to this MO-question (on contractible homogeneous spaces of Lie groups) is affirmative.
Concerning the "non-symmetric" version of the Conjecture, I am not so optimistic, so ask it as an open
Problem: Is every line metric space homeomorphic to a Euclidean space?
Added in Edit: According to the answer of Shijie Gu to this problem, the Conjecture is indeed true, so the Question has an affirmative answer. Only the Problem remains open (up to my current knowldge). Since this problem is a modification of the Busemann Conjecture which is true for Busemann G-spaces of small dimension, I expect that the answer to Problem is affirmative at least for spaces of small topological dimension, say $\le 4$.
