Uniform obstruction-size 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 $\delta$, let $\mathbb{C}_{\leq\delta}$ and $\mathbb{C}_{<\delta}$ be the corresponding class properties defined by limiting density. Uniform obstruction-size conjecture. There is a constructible function $f:\mathbb{Q}\to\mathbb{N}$ such that, for every $\delta\in\mathbb{Q}$, if $\mathcal{G}\in\operatorname{cobs}(\mathbb{C}_{\leq\delta})$ or $\mathcal{G}\in\operatorname{cobs}(\mathbb{C}_{<\delta})$, then every graph in $\operatorname{obs}(\mathcal{G})$ has at most $f(\delta)$ vertices. This is proposed as a refinement of the finite obstruction conjecture, and its status is open in the supplied text.