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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 upper half-plane, the cluster is instantaneously destroyed, i.e. all of its sites turn vacant. At the critical time $t_c := \log 2$ the process is stopped. Now 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 in the final configuration a.s. only finitely many sites in the cone have been affected by destruction.