Here is a proof of Conjecture 1.   

**Proof.**  We prove the contrapositive.  Suppose that $G$ is not a cograph.  Then $G$ has an induced subgraph $H$ such that $H \simeq P_4$.    Let $V(H)=\{1,2,3,4\}$ and $E(H)=\{12,23,34\}$.  Thus, $\mathcal{C}(H)=E(H)$ and  so $\mathcal{C}(H)^{\perp}=\{13,24\}$.  It follows that $\mathcal{C}(H)^{{\perp}{\perp}}=\{12,23,34,14\} \neq \mathcal{C}(H)$.