Vu's sharpened chromatic conjecture

Let $G$ be a graph, with maximum degree $\operatorname{\Delta}(G)$, maximum codegree $\operatorname{\Delta_2}(G)$, and chromatic number $\chi(G)$. Vu's sharpened conjecture. There is a constant $C>0$ such that

$$\chi(G)\leq \max\left\{\frac{C\operatorname{\Delta}(G)}{\log\operatorname{\Delta}(G)},\ \operatorname{\Delta_2}(G)+3\right\}.$$

This strengthens the preceding conjecture in the larger-codegree regime and proposes a bound in the smaller-codegree regime, aligning the former with Vizing's theorem. It is presented as an open sharpening.