Finite-convergence conjecture for the copositive stability-number hierarchy

Let $G$ be a graph, let $M_G$ be its graph matrix, let $F_G(x)=(x^{\circ2})^TM_Gx^{\circ2}$, and let

$$\mathcal K_n^{(r)}=\{M\in\mathcal S^n:(\sum_{i=1}^n x_i^2)^r(x^{\circ2})^TMx^{\circ2}\in\Sigma\}.$$

Define $\vartheta^{(r)}(G)=\min\{t:t(A_G+I)-J\in\mathcal K_n^{(r)}\}$. The finite-convergence conjecture. For every graph $G$, there exists $r\in\mathbb N$ such that $M_G\in\mathcal K_n^{(r)}$, equivalently $(\sum_{i=1}^n x_i^2)^rF_G\in\Sigma$, equivalently $\vartheta^{(r)}(G)=\alpha(G)$, or equivalently $\vartheta\text{-rank}(G)<\infty$. This weaker conjecture asks only for finite convergence at some order and is explicitly stated to remain open.