Stochastic domination of the range distribution by a path

Prove that, for every finite connected bipartite simple graph, the entire distribution of the range of a uniformly chosen standard height function is stochastically dominated by the range distribution for a path with the same number of vertices.

Background

The paper proves the expectation form of the Benjamini–Häggström–Mossel conjecture: among connected bipartite graphs with a fixed number of vertices, paths maximize the expected range of a uniformly chosen standard height function. A stronger assertion concerns stochastic domination of the full range distribution, rather than only its expectation.

The paper notes that stochastic domination had been established only for trees in the lazy model and spiders in the standard model. Thus the general distributional assertion for all connected bipartite graphs remains unresolved, even though the expectation inequality is proved.