On a conjecture concerning antipodal labelings for cycles

Let $G$ denote a finite, connected, simple graph, and let $D$ denote its diameter, and let $d_{G}(u, v)$ denote the distance between vertices $u$ and $v$ in $V(G)$. An antipodal labeling of $G$ is a mapping $f\colon V(G) \to \mathbb{N}_{0}$ such that, for each pair $(u, v)$ consisting of distinct ve