Non-permutation conjecture for the trivariate family $C_u$

Let $m>3$, let $u\in \mathbb{F}_{2^m}^*$, and define the function $C_u: \mathbb{F}_{2^m}^3 \to \mathbb{F}_{2^m}^3$ by

$$C_u(X,Y,Z)=(X^3+uY^2Z,Y^3+uXZ^2,Z^3+uX^2Y).$$

Non-permutation conjecture. The function $C_u$ is not a permutation and is not APN.

This conjecture concerns a trivariate representation of APN permutations in dimension $9$. The case $m=3$ is known to yield a vectorial permutation for $u\in\mathbb{F}_{2^m}\setminus\{0,1\}$, while the behavior for $m>3$ is the subject of the conjecture.