Rall's Hamiltonian-path conjecture for the domination game

Let $G$ be a graph on $n$ vertices. A Hamiltonian path is a path containing every vertex of $G$ exactly once. The game domination number $\gamma_g(G)$ is the number of moves in the domination game when Dominator starts and both players play optimally. Rall's conjecture. If $G$ has a Hamiltonian path, then

$$\gamma_g(G) \leq \left\lceil \frac{1}{2}n \right\rceil.$$

The paper proves the weaker bound $\lceil 10n/17\rceil$ for graphs with a Hamiltonian path. Thus the conjectured half-order bound remains open.