Eventual family conjecture for nested graphs of bounded algebraic co-rank

Let $\mathcal{G}=\{G_i\}_{i=1}^{\infty}$ be an infinite family of simple graphs such that $G_i$ is a proper induced subgraph of $G_j$ for all $i<j$. Eventual family conjecture. Either

$$\max\{\gamma_{\mathcal P}(G_i)\}_{i=1}^{\infty}=\infty,$$

or there exist a graph $G$, a vector $\delta\in\{0,1,-1\}^{V(G)}$, and an integer $M\in\mathbb N$ such that $G_i\in\mathcal{T}_{\delta}(G)$ for all $i\geq M$. This is presented as a weak form of the finite classification conjecture and remains open.