Do there exist a family of graphs with the property: $$\left|\alpha\left({G \boxtimes \bar{G}}\right) - \alpha\left({\overline{G \boxtimes \bar{G}}}\right)\right| = O(\log(N_{G}))$$ where $G$ is the graph, $\bar{G}$ is its complement, $\boxtimes$ denotes the strong product, $\alpha(G)$ denotes the independence number of $G$ and $N_{G}$ is the number of vertices of $G$?