De Klerk–Pasechnik conjecture on the copositive hierarchy for the stability number

Let $G$ be a non-empty graph with vertex set $V=[n]$, adjacency matrix $A_G$, stability number $4\alpha(G)4$, and associated matrix $M_G=\alpha(G)(A_G+I)-J$. Let $f_G$ be the quartic form associated with $M_G$, and let $\mathcal{K}_n^{(r)}$ denote the $r$-th sum-of-squares cone in the hierarchy. The hierarchy is defined by

$$\theta^{(r)}(G):=\min\left\{t\mid t(A_G+I)-J\in \mathcal{K}_n^{(r)}\right\}.$$

De Klerk–Pasechnik conjecture. For every non-empty graph $G$,

$$\left(\sum_{i=1}^n x_i^2\right)^{\alpha(G)-1}f_G\in \Sigma.$$

Equivalently, $M_G\in\mathcal{K}_n^{(\alpha(G)-1)}$, or $\theta^{(\alpha(G)-1)}(G)=\alpha(G)$. Reznick certification implies that the hierarchy converges to $\alpha(G)$ as $r\to\infty$, but it remains open whether $\alpha(G)-1$ steps always suffice.