Let $X$ be a topological manifold of dimension $n$, equipped with a compatible CAT(0) metric.
Are *sufficiently small* metric spheres in $X$ homeomorphic to metric spheres in Euclidean space $\mathbb{E}^n$?

[In "Ideal boundary of CAT(0) spaces" (1998) by Myung-Jin Jeon, this was unclear to the author; see the bottom of Page 104. That paper examined geodesic completeness for CAT(0) manifolds.]

**EDIT**: It is well-known that sufficiently small metric balls in any CAT(0) space are contractible. Using the manifold property, I believe that my question reduces to a very basic one:

If $U$ is a contractible open subset of $\mathbb{R}^n$, then is $U$ homeomorphic to $\mathbb{R}^n$?