List-distance consistent vertices in trees are confined to a path

A labeling of a connected graph $G$ on $n$ vertices is a bijection $c:V(G)\to\{1,\dots,n\}$; writing $c(u,v)=|c(u)-c(v)|$, a vertex $u$ is list-distance consistent if $d(u,v)<d(u,w)$ implies $c(u,v)\le c(u,w)$ for all $v,w$. The maximum number of such vertices over all labelings is the list-distance