Vu's codegree conjecture for chromatic number

Let $G$ be a graph. Write $\operatorname{\Delta_2}(G)$ for its maximum codegree and $\operatorname{\Delta}(G)$ for its maximum degree. Vu's conjecture. Fix $\varepsilon_1,\varepsilon_2>0$. If

$$\operatorname{\Delta_2}(G)\geq \varepsilon_1\operatorname{\Delta}(G),$$

then

$$\chi(G)\leq \operatorname{\Delta_2}(G)+\varepsilon_2\operatorname{\Delta_2}(G),$$

provided $\operatorname{\Delta}(G)$ is sufficiently large. Vu proposed this in 2002, originally for the stronger list chromatic number. The paper proves a bound within $3$ of the maximum codegree for claw-free graphs, but the general conjecture remains open.