 the following weaker condition:  0.  There exists a dynamically impossible set $N$ for the intensity measure $\nu$ such that $$\lim_{\varepsilon \rightarrow 0^+} \Delta \tilde{E}^{H^\varepsilon}_\eta (\gamma_x) = \Delta \tilde{E}^{H^0}_\eta (\gamma_x)$$ for all $\eta \in \mathcal{N}(S \times G)$ and $\gamma_x \in S \times G$ satisfying $\eta, \eta + \delta_{\gamma_x} \in N^c$.  Applications {#