Domination bound for complementary self-centered graphs

Let $G$ be a graph of order $n$. Write $\overline{G}$ for its complement, and let $\gamma(G)$ denote the domination number of $G$. A graph is in $SCC(2)$ when both $G$ and $\overline{G}$ are self-centered and

$$\operatorname{rad}(G)=\operatorname{diam}(G)=\operatorname{rad}(\overline{G})=\operatorname{diam}(\overline{G})=2.$$

Domination conjecture for $SCC(2)$ graphs. If $G\in SCC(2)$, then

$$\gamma(G)\leq\left\lceil\sqrt{n}\right\rceil$$

and

$$\gamma(\overline{G})\leq\left\lceil\sqrt{n}\right\rceil.$$

The paper introduces this conjecture for the exceptional class left after proving polynomial-time decidability outside $SCC(2)$. It would give simultaneous domination-number bounds for a graph and its complement in the unresolved case.