The five-holed graph chromatic bound conjecture

A graph $G$ is $5$-holed if every induced cycle of $G$ of length at least four has length exactly five. Let $\chi(G)$ and $\omega(G)$ denote the chromatic and clique numbers of $G$.

Five-holed graph chromatic bound conjecture. If $G$ is $5$-holed, then

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

The paper proves the analogous bound for odd $\ell\geq 7$ and proposes this as the remaining five-holed case; the source gives no resolution of the conjecture.