  Hermon and Hutchcroft have recently proved the long-standing conjecture that
in Bernoulli(p) bond percolation on any nonamenable transitive graph G, at any
p > p_c(G), the probability that the cluster of the origin is finite but has a
large volume n decays exponentially in n. A corollary is that all infinite
clusters have anchored expansion almost surely. They have asked if these
results could hold more generally, for any finite energy ergodic invariant
percolation. We give a counterexample, an invariant percolation on the
4-regular tree.
