The additive 1/2-conjecture for the domination game

Let $G$ be a graph on $n$ vertices with minimum degree at least $2$. The game domination number $\gamma_g(G)$ is the number of moves in the domination game when Dominator starts and both players play optimally. Bujtás–Iršič–Klavžar's additive conjecture. There exists a universal constant $C$ such that

$$\gamma_g(G) \leq \frac{1}{2}n+C.$$

This is a relaxation of the conjectured bound $\gamma_g(G) \leq \lceil n/2\rceil$ for graphs of minimum degree at least $2$. The paper's $10n/17+1/17$ result does not reach the coefficient $1/2$, so this conjecture remains open.