$  0.  $\lim_{\varepsilon \rightarrow 0^+} \Delta \tilde{E}^{H^\varepsilon}_\eta (\gamma_x) = \Delta \tilde{E}^{H^0}_\eta (\gamma_x)$ for every $\eta \in \mathcal{N}(S \times G)$ and $\gamma_x \in S \times G$.  Then 