The two-thirds conjecture for locating-total dominating sets

Let $G$ be a graph of order $n$. A set $D\subseteq V(G)$ is a locating-total dominating set if every vertex has a neighbor in $D$, and any two vertices outside $D$ have distinct neighborhoods within $D$. Write $\gamma_t^L(G)$ for the minimum size of such a set. A graph is twin-free if it contains no pair of twins, where twins have either the same open neighborhood or the same closed neighborhood; it is isolate-free if it has no isolated vertex.

Two-thirds conjecture. Every twin-free isolate-free graph $G$ of order $n$ satisfies

$$\gamma_t^L(G)\leq\frac{2n}{3}.$$

The conjecture asks for a universal upper bound on locating-total domination in the absence of twins. It was proved for graphs with no $4$-cycle, for line graphs, and for graphs of sufficiently large minimum degree under a related conjecture; a stronger bound with factor $\frac{1}{2}$ is known for claw-free cubic graphs. The general case remains open.