Gyárfás' boundedness conjecture for graphs with odd-cycle chromatic number at most three

Let $G$ be a graph, and let $r(G)$ denote the maximum chromatic number of a subgraph spanned by an odd cycle of $G$, while $\chi(G)$ denotes the chromatic number of $G$. Gyárfás' boundedness conjecture. There exists a constant $k$ (perhaps $k=4$) such that if $G$ is a graph with $r(G)\leq 3$, then $\chi(G)\leq k$. This is the main problem discussed in the paper and is attributed to Gyárfás; the best progress cited is the logarithmic upper bound of Randerath and Schiermeyer, and the conjecture remains open.