Finite-convergence conjecture for the theta hierarchy of a graph

Let $G$ be a graph, let $\alpha(G)$ denote its stability number, and let $\vartheta^{(r)}(G)$ denote the level-$r$ semidefinite parameter in the theta hierarchy.

Finite-convergence conjecture. For any graph $G$,

$$\vartheta^{(r)}(G)=\alpha(G)\quad\text{for some }r\in\mathbb{N}.$$

This is a weaker conjecture than the De Klerk–Pasechnik conjecture: it asks only for convergence at some finite level, without specifying the level. Finite convergence at any step is not known in general.