Conjecture on the center and median of a maximum-Wiener-index tree

Let $\mathfrak{T}_n(d)$ be the family of trees on $n$ vertices with diameter $d$, let $W(T)$ denote the Wiener index of a tree $T$, let $C(T)$ denote its center, and let $M(T)$ denote its median. Choose a tree $T_0\in\mathfrak{T}_n(d)$ satisfying

$$W(T_0)=\max\{W(T):T\in\mathfrak{T}_n(d)\}.$$

Center–median conjecture. The center and median of $T_0$ have distance zero:

$$d(C(T_0),M(T_0))=0.$$

This proposes an additional structural restriction on trees attaining the maximum Wiener index among trees with fixed order and diameter. The source presents it as an unresolved conjecture intended to narrow the possible extremal trees.