Variance-optimal functions on trees are centered at a vertex

Let $T$ be a tree and let $X$ be a variance-optimal function over $T$. For vertices $u,r\in V(T)$, write $d(u,r)$ for their graph distance.

Tree characterization conjecture. There exists a vertex $r$ such that, for every vertex $u\in V(T)$,

$$|X(u)-X(r)|=d(u,r).$$

Moreover, if $T$ is not a path, then $r$ may be chosen so that $d(r)\geq 3$.

The conjecture is proposed as a simple characterization of variance-optimal functions on trees and as a strengthening of the preceding structural result about hairs. The supplied text gives no resolution.