Once we considered a similar problem but around infinity, try to look in our paper "Asymptotical flatness and cone structure at infinity".
If my intuition is correct, it is possible to prove in all dimensions except 4. In dimension 4 there there might be the following funny example (I did not really check it, just an intuition). Your singular point has tangent space $\mathbb R^3$. the $r$-spheres around this point are Berger spheres, so its curvature is very much like curvature of $r(S^2\times \mathbb R)$, the size of Hopf fibers goes to $0$ very fast say as $r^{10}$. EDIT: I checked this "example"; it does not really work, but it seems to make problem for our method.
Hope it helps, if not ask me again, I'm sure we can cook something out :)
