Huang–Zhou–Chang's 5/4 coloring conjecture for even-hole-free graphs

A hole is an induced cycle of length at least four, and a graph is even-hole-free if it has no hole of even length. For a graph $G$, let $\chi(G)$ denote its chromatic number and $\omega(G)$ its clique number. Huang–Zhou–Chang's conjecture. Every even-hole-free graph $G$ satisfies

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

Every even-hole-free graph is known to satisfy $\chi(G)\leq 2\omega(G)-1$, but whether that bound is best possible remains open. The conjecture proposes the sharper linear coloring bound above.