Minor-closed-family conjecture for oriented chromatic number

For a minor-closed family of graphs $\mathcal{G}$, let $\omega_o(\mathcal{G})$ be the order of a largest oriented clique in $\mathcal{G}$.

Minor-closed-family conjecture. There exists a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $f(n)=n^{1+o(1)}$ and, for every minor-closed family of graphs $\mathcal{G}$,

$$\max_{G\in\mathcal{G}} \operatorname{\chi_o}(G) \leq f(\omega_o(\mathcal{G})).$$

The conjecture proposes that, across every minor-closed graph family, the maximum oriented chromatic number is at most a nearly linear function of the largest oriented clique order. The paper presents this as a possible general pattern motivated by the bounds for graphs on surfaces.