Scott's odd induced subgraph conjecture

Let $G$ be a finite simple graph, let $f_o(G)$ be the maximum order of an induced subgraph of $G$ whose every vertex has odd degree, and let $\chi(G)$ be the chromatic number of $G$. Assume that $G$ has no isolated vertices. Scott's conjecture.

$$f_o(G)\geq \frac{|V(G)|}{\chi(G)}.$$

Scott's conjecture strengthens his proven bound by removing the factor $2$. The conjecture is a central chromatic formulation of the problem of finding large odd induced subgraphs; the supplied text gives no resolution status for this claim.