Non-liftability of wild z-automorphisms from polynomial to free associative algebras

Let $K$ be a field of characteristic $0$, let $K[x,y,z]$ be the commutative polynomial algebra, and let $K\langle x,y,z\rangle$ be the free associative algebra. An automorphism is wild if it is not tame, and a $z$-automorphism if it fixes $z$. Non-liftability conjecture. If $\varphi$ is a wild $z$-automorphism of $K[x,y,z]$, then it cannot be lifted to a $z$-automorphism, or to any automorphism, of $K\langle x,y,z\rangle$. The claim concerns the relationship between wild automorphisms of the commutative polynomial algebra and automorphisms of the corresponding free associative algebra; its resolution is not indicated in the supplied text.