Linear oriented chromatic number for bounded-degeneracy graphs

For an integer $d\geq 1$, let $\chi_o(G)$ denote the oriented chromatic number and $\chi_2(G)$ the $2$-dipath chromatic number of a graph $G$.

Bounded-degeneracy linearity conjecture. For every integer $d\geq 1$, there exists a constant $c$ depending on $d$ such that

$$\chi_o(G)\leq c\chi_2(G)$$

for every graph $G$ with degeneracy at most $d$.

The conjecture asks whether the upper bound for the oriented chromatic number in terms of the $2$-dipath chromatic number can be made linear on every fixed-degeneracy graph class. Its status is not specified in the supplied text.