p\] For any bounded set $\mathcal{K} \subseteq \mathcal{D}_e^*$ at a positive distance from ${\partial}\mathcal{D}_e^*$ and $\delta > 0$ there exists a constant $C > 0$ such that $$\sup_{u \in \mathcal{K}} P_u ( |\tau_\varepsilon - \tau| > \delta ) \leq e^{- \frac{C}{\varepsilon^0}}.$$  