Non-APN and non-permutation conjecture for the family $C_u$

Let $m>3$ and let $uein \mathbb{F}_{2^m}$. Define

$$\phi_u\colon (\mathbb{F}_{2^m})^3\to \mathbb{F}_{2^m},\qquad \phi_u(x,y,z)=x^3+uy^2z,$$

and

$$C_u(x,y,z)=\bigl(\phi_u(x,y,z),\phi_u(y,z,x),\phi_u(z,x,y)\bigr).$$

Non-APN and non-permutation conjecture. The function $C_u$ is not APN. Further, if $u\neq 0$, then $C_u$ is not a permutation.

The case $m=3$ is exceptional, since the family contains APN permutations there. The authors report computational evidence that the conjectured non-permutation statement holds for several odd dimensions, while the APN question remains open for all $m>3$.