The closed neighborhood conflict-free chromatic number of a graph $G$, denoted by $χ_{CN}(G)$, is the minimum number of colors required to color the vertices of $G$ such that for every vertex, there is a color that appears exactly once in its closed neighborhood. Pach and Tardos [Combin. Probab. Comput. 2009] showed that $χ_{CN}(G) = O(\log^{2+\varepsilon} Δ)$, for any $\varepsilon > 0$, where $Δ$ is the maximum degree. In [Combin. Probab. Comput. 2014], Glebov, Szabó and Tardos showed existence of graphs $G$ with $χ_{CN}(G) = Ω(\log^2Δ)$. In this paper, we bridge the gap between the two bounds by showing that $χ_{CN}(G) = O(\log^2 Δ)$.