Connectivity bound for graphs exceeding Reed's bound

Let $G$ be a graph, with chromatic number $\chi$, clique number $\omega$, maximum degree $\Delta$, order $n$, and let $\overline{\kappa}=\kappa(\overline{G})$ denote the connectivity of its complement. The connectivity bound conjecture. There exists a constant $C>0$ such that

$$\chi>\left\lceil\frac{1}{2}(\omega+\Delta+1)\right\rceil\Rightarrow\overline{\kappa}\geq C\log(n).$$

This is intended to express that a graph violating Reed's conjectural upper bound must have a highly connected complement; the supplied source gives no resolution status.