Finite obstruction conjecture for classes of a fixed limiting density

For a graph class $\mathcal{G}$, let $\operatorname{obs}(\mathcal{G})$ denote its minor obstruction set. For a class property $\mathbb{C}$, let $\operatorname{cobs}(\mathbb{C})$ be the set of inclusion-minimal proper minor-closed graph classes not in $\mathbb{C}$. For rational $\delta$, write $\mathbb{C}_{\leq\delta}$ and $\mathbb{C}_{<\delta}$ for the corresponding class properties defined by limiting density. Finite obstruction conjecture. For every $\delta\in\mathbb{Q}$, both $\operatorname{cobs}(\mathbb{C}_{\leq\delta})$ and $\operatorname{cobs}(\mathbb{C}_{<\delta})$ are finite. The paper establishes the relevant characterization below $3/2$ and presents finiteness for every rational density as an open research direction; the conjecture is also described as a special instance of the $\omega^2$-well-quasi-ordering conjecture.