Erdős, Gimbel and Straight's conjecture for graphs with clique number less than five

Let $G$ be a graph, and write $\omega(G)$ for its clique number, $\zeta(G)$ for its cochromatic number, and $\chi(G)$ for its chromatic number. Erdős, Gimbel and Straight's conjecture. Every graph $G$ with $\omega(G)<5$ and $\zeta(G)>3$ satisfies

$$\chi(G)\le \zeta(G)+2.$$

Erdős, Gimbel and Straight found examples showing that the corresponding bound cannot generally be reduced below $3$ for graphs with clique number less than five, but their examples all have cochromatic number at most $3$. The conjecture asserts that graphs with cochromatic number greater than $3$ obey the sharper bound; it remains open.