 that for every $t_0 > 0$ $$\lim_{\varepsilon \rightarrow 0}\left[ \sup_{u,v \in B_{\rho}} \left[ \sup_{t > t_0} |P_u(\tau_{\varepsilon}(\partial G)  > t\gamma_{\varepsilon}) - P_{v}(\tau_{\varepsilon}(\partial G) > t\gamma_{\varepsilon})|\right]\right] = 0.$$ 