The three-sevenths upper-bound conjecture for the game isolation number

Let $G$ be a graph, let $n(G)$ denote its order, and let $\iota_{\rm g}(G)$ and $\iota_{\rm g}'(G)$ denote its game isolation number and Staller-start game isolation number, respectively. A graph has no $K_2$-components when none of its connected components is isomorphic to $K_2$.

Three-sevenths upper-bound conjecture. For any graph $G$ with no $K_2$-components,

$$\iota_{\rm g}(G) \leq \left\lceil\frac{3}{7}n(G)\right\rceil \quad\text{and}\quad \iota_{\rm g}'(G) \leq \left\lceil\frac{3}{7}n(G)\right\rceil.$$

The paper presents this as a conjecture from earlier work and studies refinements and examples showing that the three-sevenths bound is sharp for suitable graph families. Its resolution is not stated in the supplied text.