The bunkbed conjecture holds in the p ↑ 1 limit

Let G = (V, E) be a countable graph. The Bunkbed graph of G is the product graph G x K₂, which has vertex set V x {0,1} with "horizontal" edges inherited from G and additional "vertical" edges connecting (w,0) and (w,1) for each w ϵ V. Kasteleyn's Bunkbed conjecture states that for each u, v ϵ V and p ϵ [0,1], the vertex (u,0) is at least as likely to be connected to (v,0) as to (v,1) under Bernoulli-p bond percolation on the bunkbed graph. We prove that the conjecture holds in the p ↑ 1 limit in the sense that for each finite graph G there exists ε(G) > 0 such that the bunkbed conjecture holds for p ⩾ 1 - ε(G).