Cohen–Macaulayness conjecture for Rees algebras of complementary edge ideals

Let $G$ be a finite simple graph, and let $I_c(G)$ denote its complementary edge ideal in the polynomial ring associated to $G$. Write $\mathcal{R}(I_c(G))$ for the Rees algebra of $I_c(G)$. Cohen–Macaulayness conjecture. The Rees algebra

$$\mathcal{R}(I_c(G))$$

is Cohen–Macaulay. This conjecture is motivated by the computed depth behavior of powers of complementary edge ideals and by the relationship between Cohen–Macaulayness of the Rees algebra and the asymptotic depth of those powers; its general status is not resolved in the supplied source.