A bound for $[\mathbb{C}(x,y,z):\mathbb{C}(p,q,r)]$, where $\operatorname{Jac}(p,q,r) \in \mathbb{C}^{\times}$

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}$ .

Any hints and comments are welcome! Thank you.