Upper bounds for inverse domination in graphs

In any graph $G$, the domination number $\gamma(G)$ is at most independence $\alpha(G)$. The \emph{Inverse Domination Conjecture} says that, in isolate-free there exists pair of vertex-disjoint dominating sets $D, D'$ with $|D|=\gamma(G)$ and $|D'| \leq \alpha(G)$. Here we prove that this statement true if upper bound $\alpha(G)$ replaced by $\frac{3}{2}\alpha(G) - 1$ (and $G$ not a clique). We also conjecture holds whenever $\gamma(G)\leq 5$ or $|V(G)|\leq 16$.