Tightness for the Cover Time of the two dimensional sphere

Let $C_{ϵ,S^2}^{\ast }$ denote the cover time of the two dimensional sphere by a Wiener sausage of radius $ϵ$. We prove that $$\sqrt{C_{ϵ,S^2}^{\ast }}-\sqrt{\frac{2A_{S^2}}{\pi }}(\log {ϵ}^{-1}-\frac{1}{4}\log \log {ϵ}^{-1})$$ is tight, where $A_{S^2}=4\pi$ denotes the Riemannian area of $S^2$.