Bounding clique size in squares of planar graphs

Wegner conjectured that if $G$ is a planar graph with maximum degree $\Delta \ge 8$, then $\chi \left(G^2\right)\le \left(\frac{3}{2}\Delta \right)+1$. This problem has received much attention, but remains open for all $\Delta \ge 8$. Here we prove an analogous bound on $\omega \left(G^2\right)$: If $G$ is a plane graph with $\Delta \left(G\right)\ge 36$, then $\omega \left(G^2\right)\le \lfloor \frac{3}{2}\Delta \left(G\right)\rfloor +1$. In fact, this is a corollary of the following lemma, which is our main result. If $G$ is a plane graph with $\Delta \left(G\right)\ge 19$ and $S$ is a maximal clique in $G^2$ with $\left|S\right|\ge \Delta \left(G\right)+20$, then there exist $x,y,z\in V\left(G\right)$ such that $S=\{w:|N\left[w\right]\cap \{x,y,z\}|\ge 2\}$.