By Kuratowski's theorem, every nonplanar graph contains a (topological) minor of $K_5$ or $K_{3,3}$.
But I observed that every time I construct a $4$-connected nonplanar graph, it always contains not only a $K_{3,3}$-minor but also a $K_5$-minor.
Moreover, although I tried many times, I cannot construct a $4$-connected nonplanar graph containing only $K_{3,3}$-minors!
So I want to know whether the following statement is true:

Every $4$-connected nonplanar graph contains a $K_5$-minor.

Unfortunately, I could not find any references about this topic.
If the statement is true, can you give me a proof?
Or if it's not, can you show me a counterexample?
Thanks a lot.