Exponential estimates conjecture for the contact process in dynamic percolation

Let $(\mathbf{C},\mathbf{B})$ be a contact process in dynamic percolation (CPDP) with rates $\lambda,r,\alpha,\beta>0$ on the $d$-dimensional integer lattice. Let $\theta_{\mathrm{DP}}(\lambda,r,\alpha,\beta)$ denote its survival probability.

Exponential estimates conjecture. If

$$\theta_{\mathrm{DP}}(\lambda,r,\alpha,\beta)>0,$$

then there exist constants $C_1,C_2,M>0$ such that the three conditions labelled $\mathrm{ConjectureEq1}$, $\mathrm{ConjectureEq2}$ and $\mathrm{ConjectureEq3}$ in the source are fulfilled.

These estimates are the hypotheses used to derive the asymptotic shape theorem. Comparable estimates have been proved for the contact process in a static random environment and for the contact process with ageing; proving them for the CPDP would extend those methods to this dynamical setting.