Properness of the hierarchy of graph classes $\mathcal{G}_k$

Let $\mathcal{G}_k$ be the graph class defined in the paper for each $k\in\mathbb{N}$. Hierarchy properness conjecture. For each $k\in\mathbb{N}$, the inclusion

$$\mathcal{G}_k\subseteq\mathcal{G}_{k+1}$$

is proper. The conjecture asserts that the hierarchy of these graph classes never collapses; the supplied text gives no resolution status beyond the claim itself.