The closed neighborhood conflict-free chromatic number of a graph $G$, denoted by $\chi_{CN}(G)$, 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 showed that $\chi_{CN}(G) = O(\log^{2 + \varepsilon} \Delta)$, for any $\varepsilon > 0$, where $\Delta$ is the maximum degree. In 2014, Glebov et al. showed existence of graphs $G$ with $\chi_{CN}(G) = \Omega(\log^{2} \Delta)$. In this article, we bridge the gap between the two bounds by showing that $\chi_{CN}(G) = O(\log^{2} \Delta)$.