The 5/4 chromatic bound conjecture for even-hole-free graphs

Let $G$ be an even-hole-free graph, meaning that $G$ has no induced cycle of even length at least four. Write $\chi(G)$ for its chromatic number and $\omega(G)$ for its clique number. The 5/4 bound conjecture. For every even-hole-free graph $G$,

$$\chi(G)\leq \left\lceil \frac{5}{4}\omega(G) \right\rceil.$$

The paper proves the same bound for the subclass of $(P_7,$ even-hole$)$-free graphs and notes that equal-size blowups of $C_5$ suggest optimality, but the full even-hole-free case remains open.