The half-order conjecture for locating-dominating sets in twin-free graphs

Let $G$ be a finite graph. A set $C\subseteq V(G)$ is a locating-dominating set if it dominates every vertex outside $C$ and, for every two distinct vertices $u,v\in V(G)\setminus C$, the sets $N(u)\cap C$ and $N(v)\cap C$ are distinct. Write $\gamma^{LD}(G)$ for the minimum cardinality of a locating-dominating set, and let $n=|V(G)|$ be the order of $G$. A graph is twin-free if it has neither two vertices with equal open neighborhoods nor two vertices with equal closed neighborhoods; an isolated vertex has degree zero. The half-order conjecture. Every twin-free graph $G$ of order $n$ without isolated vertices satisfies

$$\gamma^{LD}(G)\leq \frac{n}{2}.$$

This conjecture, originally proposed in the cited work and subsequently studied in the cited formulation, asserts a sharp universal upper bound for locating-dominating sets in twin-free graphs. Its resolution is not specified in the supplied source material.