Membership conjecture for graph matrices in the copositive hierarchy

Let $G$ be a graph with stability number $\alpha(G)$, and let $A_G$, $I$, and $J$ denote its adjacency, identity, and all-ones matrices, respectively. Define

$$M_G=\alpha(G)(A_G+I)-J.$$

By the Motzkin–Straus theorem, $M_G$ is copositive. Graph-matrix hierarchy membership conjecture. For every graph $G$,

$$M_G\in\bigcup_{r\geq 0}{\mathcal K}_n^{(r)}.$$

This weaker conjecture asks whether every graph matrix eventually appears in the sum-of-squares copositive hierarchy. The source states that it remains open, and notes that even this eventual membership is not known in general.