Positive critical regrowth parameter for self-organized forest fires

Let $p_c$ denote the critical probability for site percolation on the square lattice. For a positive integer $n$, set $B=[0,4n]\times[0,3n]$ and $A=[n,3n]\times[n,2n]$. Starting from site percolation on $B$ with density $p_c$, destroy every occupied cluster connected to the boundary $\partial B$, then independently occupy each resulting vacant site with probability $\delta$. If $p_n(\delta)$ is the probability that the resulting configuration has an occupied vertical crossing of $A$, define

$$\hat{\delta}_c=\sup\{\delta:p_n(\delta)\text{ is bounded away from }1\text{ uniformly in }n\}.$$

Positive critical regrowth conjecture. The critical regrowth parameter is positive:

$$\hat{\delta}_c>0.$$

This asserts that the spatial dependencies created by destroying the boundary-connected occupied cluster do not make the bulk essentially supercritical for every positive regrowth probability. The source does not provide a resolution of this claim.