If there is a convergence action on the space $X = R^{n+1}\cup S^{n}$, is it true that X is a closed ball?

Question

Let G be a group that acts on a Hausdorff compact space X with the convergence property such that the limit set Y is homeomorphic to the sphere $S^{n}$ and X-Y is homeomorphic to $\mathbb{R}^{n+1}$. Is it true that X is homeomorphic to the closed ball of dimension n+1?

I'm pretty sure that I already saw something like this somewhere (I don't know if it works for every dimension) but I have no idea where it was.