Problem. Let $G$ be a graph with a $K_5$ minor. Prove that $G$ contains either a $K_5$ or a $K_{3,3}$ topological minor.
I'm having a hard time believing this result. Consider the graph $G$ obtained from $K_5$ by replacing one of its vertices with a cycle of length 4:
Where is the $K_5$ or $K_{3,3}$ topological minor?