The Anick wildness conjecture

Let $K\langle x,y,z\rangle$ be the free associative algebra over a field $K$, and define

$$\nu_1=(x+z(xz-zy),y+(xz-zy)z,z).$$

The Anick wildness conjecture. The automorphism $\nu_1$ of $K\langle x,y,z\rangle$ is wild. The conjecture was later proved by Umirbaev, who established that the Anick automorphism is wild even in the free metabelian algebra and hence in the free associative algebra.