Self-destructive percolation

Consider ordinary site percolation on an infinite graph in which the sites, independent of each other, are occupied with probability p and vacant with probability 1 - p. Now suppose that, by some "catastrophe," all sites which are in an infinite occupied cluster become vacant. Finally, each vacant site gets an extra enhancement to become occupied. More precisely, each site that was already vacant or that was made vacant by the catastrophe, becomes occupied with probability S (independent of the other sites). When p is larger than but close to the critical value p _c one might believe (for "nice" graphs) that only a small δ is needed to have an infinite occupied cluster in the final configuration. This appears to be indeed the case for the binary tree. However, on the square lattice we strongly conjecture that this is not true. We discuss the background for these problems and also show that the conjecture, if true, has some remarkable consequences.