Neighboring clusters at Bernoulli percolation

The study of Bernoulli percolation on general infinite transitive graphs was initiated by a paper of Benjamini and Schramm in 1996, and has been intensive since then. One of the interesting phenomena is that for certain graphs there is a value of p when there are infinitely many infinite components. This is conjectured to be a characterizing property of nonamenable graphs. Haggstrom, Peres and Schonmann asked whether it can happen that two infinite components of such a percolation come at distance 1 from each other at infinitely many places. We give a negative answer to this question.