Critical heights of destruction for a forest-fire model on the half-plane

Consider the following forest-fire model on the upper half-plane of the triangular lattice: Each site can be "vacant" or "occupied by a tree". At time 0 all sites are vacant. Then the process is governed by the following random dynamics: Trees grow at rate 1, independently for all sites. If an occupied cluster reaches the boundary of the half-plane or if it is about to become infinite, the cluster is instantaneously destroyed, i.e. all of its sites turn vacant. Let t(c) = log 2 denote the critical time after which an infinite cluster first appears in the corresponding pure growth process, where there is only the growth of trees but no destruction mechanism. Choose an arbitrary infinite cone in the half-plane whose apex lies on the boundary of the half-plane and whose boundary lines are non-horizontal. We prove that at time t(c) almost surely only finitely many sites inside the cone have been affected by destruction in the forest-fire process.