Y. Zhang (in his PhD thesis) and P. I. Katsylo proved the following nice result; the two proofs are different, see: Zhang's thesis and Katsylo's paper:

Let $f: (x,y) \mapsto (p,q)$ be a $k$-algebra endomorphism of $\mathbb{C}[x,y]$ having an invertible Jacobian, namely, $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. Then the degree of the field extension $\mathbb{C}(p,q) \subseteq \mathbb{C}(x,y)$ is $\leq \min{ \{\deg(p),\deg(q)\}}$.

Now let $f: (x,y,z) \mapsto (p,q,r)$ be a $k$-algebra endomorphism of $\mathbb{C}[x,y,z]$ having an invertible Jacobian, namely, $\operatorname{Jac}(p,q,r) \in \mathbb{C}-\{0\}$.

Is the following claim true?

The degree of the field extension $\mathbb{C}(p,q,r) \subseteq \mathbb{C}(x,y,z)$ is $\leq (\min{ \{\deg(p),\deg(q),\deg(r)\})^2}$.

**Remark:** I have tried to generalize Zhang's proof to higher dimensions (or at least to dimension three), but I have a few gaps (I have tried to mimick his proof).
I do not know how to generalize Katsylo's proof to dimension three.

Any hints and comments are welcome! Thank you.