Finite classification conjecture for graphs of bounded algebraic co-rank

For $k\geq1$, let

$$\Gamma_{\leq k}=\{G: G\text{ is a simple connected graph with }\gamma(G)\leq k\},$$

and let $\mathcal{T}_{\delta}^*(G)$ denote the set of induced subgraphs of one graph in $\mathcal{T}_{\delta}(G)$. Finite classification conjecture. For every $k\geq1$,

$$\Gamma_{\leq k}=\bigcup_{(G,\delta)\in\mathcal{G}}\mathcal{T}_{\delta}^*(G)$$

for some finite set $\mathcal{G}$ of pairs $(G,\delta)$, where $G$ is a simple graph and $\delta\in\{0,1,-1\}^{V(G)}$. This extends the known classifications for algebraic co-rank at most two and three; the general finite classification remains conjectural.