Critical extinction conjecture for the contact process in a dynamic percolation

Let $(\mathbf{C},\mathbf{B})$ be the contact process in dynamic percolation (CPDP), with infection rate $\lambda$, recovery rate $r$, and initial configuration $(C,B)$, where $C$ is a finite nonempty set of infected vertices. Write $\lambda_c(r,C,B)$ for its critical infection rate and $\lambda_c^{\pi}(r)$ for the critical infection rate when the background starts in its stationary distribution.

Critical extinction conjecture. For every finite nonempty $C$ and every admissible background configuration $B$, one has

$$\lambda_c(r,C,B)=\lambda_c^{\pi}(r).$$

This conjecture asserts independence of the critical infection rate from the initial configuration in the CPDP, including at criticality. The preceding results establish independence under a growth condition and for the interior of the survival region; the conjecture concerns the remaining boundary case.