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The famous Wegner's Planar Graph Conjecture asserts tight upper bounds on thechromatic number of the square $G^2$ of a planar graph $G$, depending on themaximum degree $\Delta(G)$ of $G$. The only case that the conjecture isresolved is when $\Delta(G)=3$, which was proven to be true by Thomassen, andindependently by Hartke, Jahanbekam, and Thomas. For $\Delta(G)=4$, Wegner'sPlanar Graph Conjecture states that the chromatic number of $G^2$ is at most 9;even this case is still widely open, and very recently Bousquet, de Meyer,Deschamps, and Pierron claimed an upper bound of 12.  We take a completely different approach, and show that a relaxation ofproperly coloring the square of a planar graph $G$ with $\Delta(G)=4$ can beachieved with 9 colors. Instead of requiring every color in the neighborhood ofa vertex to be unique, which is equivalent to a proper coloring of $G^2$, weseek a proper coloring of $G$ such that at most one color is allowed to berepeated in the neighborhood of a vertex of degree 4, but nowhere else.

Relaxation of Wegner's Planar Graph Conjecture for maximum degree 4