Rall's 1/2-conjecture on the domination game

Let $G$ be a graph, and let $n(G)$ denote its order. The graph $G$ is traceable if it contains a Hamiltonian path, and $\gamma_g(G)$ denotes its game domination number in the Dominator-start game.

Rall's 1/2-conjecture. If $G$ is traceable, then

$$\gamma_g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil.$$

The conjecture was proposed by D. Rall and first published by James et al. The paper proves it for unicyclic graphs and establishes it for several additional graph families, but the general case remains open.