Square-coloring codegree conjecture

Let $G$ be a graph. Its square $G^2$ is obtained by joining distinct vertices that are connected by a two-edge path in $G$. Let $\operatorname{\Delta}(G)$ be the maximum degree, $\operatorname{\Delta_2}(G)$ the maximum codegree, and $\chi(G^2)$ the chromatic number of the square. Square-coloring conjecture. There is a constant $C>0$ such that

$$\chi(G^2)\leq \max\left\{\frac{C\operatorname{\Delta}(G)^2}{\log\operatorname{\Delta}(G)},\ \frac{5}{4}\bigl(\operatorname{\Delta_2}(G)+2\bigr)^2\right\}.$$

This is an analogue of Vu's conjecture for coloring graph squares and is proposed as an open problem.