Exponential lower-bound conjecture for the oriented chromatic number of degenerate graphs

Let $d\text{-degenerate}$ graphs be graphs in which every subgraph has a vertex of degree at most $d$, and let $\chi_o(G)$ denote the oriented chromatic number of a graph $G$. For a positive integer $n$, let $G$ be a graph on $n$ vertices. Exponential lower-bound conjecture. There exists a constant $c>0$ such that for all $d\geq 2$, there are infinitely many integers $n$, and $d$-degenerate graphs $G$ on $n$ vertices with

$$\chi_o(G) \geq c 2^{d}\sqrt{n}.$$

The conjecture asserts that the exponential dependence on the degeneracy parameter $d$ is necessary for lower bounds on the oriented chromatic number. The preceding discussion motivates this through comprehensive graphs, but the proposed lower bound and the requested novel construction remain open in the source.