  We study tree-indexed random walks as introduced by Benjamini, H\"aggstr\"om,
and Mossel, i.e. labelings of a tree for which adjacent vertices have labels
differing by 1. It is a conjecture of those authors that the distribution of
the range for any such tree is dominated by that of a path on the same number
of edges. The two main variants of this conjecture considered in the literature
are the $\textit{standard}$ walks, in which adjacent vertices must have labels
differing by $\textit{exactly}$ 1, and $\textit{lazy}$ walks, in which adjacent
vertices must have labels differing by $\textit{at most}$ 1. We confirm this
conjecture for all trees in the lazy case and provide some partial results in
the standard case.
