fedja was right to guess that I had an idea to solve the problem. Reading that Yuval Peres is working on it scared me into writing something down as quickly as possible. Any comments on my manuscript (below) are very welcome.
http://w3.impa.br/~rimfo/artigos_local/coalescing_submittedTAMS_03092010.pdfhttp://arxiv.org/abs/1009.0664