Conjecture that chromatic number is not bounded by a constant multiple of r(G)

Let $r(G)$ be the maximum chromatic number of a subgraph spanned by an odd cycle of a graph $G$, and let $\chi(G)$ be the chromatic number of $G$. Linear separation conjecture. For every constant $c>0$ there exists a graph $G$ with

$$\chi(G)>c\cdot r(G).$$

The source proposes this as an improvement of its main lower-bound result. It is stated as a conjecture for future work, and no resolution is given.