Diameter Pairs 1750 pts

Let $G$ be a simple connected undirected graph on $n$ vertices. Write $\operatorname{dist}(u,v)$ for the minimum number of edges in a path from $u$ to $v$, and let $D$ be the maximum distance between two vertices. Assume

$$2D>n.$$

Let $a,b,c,d$ be four distinct vertices satisfying

$$\operatorname{dist}(a,b)=\operatorname{dist}(c,d)=D.$$

Prove that at least one of the following holds:

$$\begin{aligned} &\operatorname{dist}(a,c)=\operatorname{dist}(b,d)=D,\\ &\text{or}\quad\operatorname{dist}(a,d)=\operatorname{dist}(b,c)=D. \end{aligned}$$