Does every $4$-connected nonplanar graph contain a $K_5$-minor?

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.

Yes, this is true and follows from Wagner's theorem. Wagner's theorem asserts that every graph with no $$K_5$$ minor can be built from $$0$$-, $$1$$-, $$2$$-, and $$3$$-sums from planar graphs and a fixed $$8$$ vertex non-planar graph called the Wagner graph. Since the Wagner graph is not $$4$$-connected, this implies that every $$4$$-connected graph with no $$K_5$$ minor is planar.
Alternatively, there is a short proof provided you are happy to assume Kuratowski's theorem. The proof idea is to start with a model of $$K_{3,3}$$ and then use $$4$$-connectivity to 'augment' the model of $$K_{3,3}$$ to a model of $$K_5$$. See the paper A Quick Proof of Wagner's Equivalence Theorem by Young.