Non-APN conjecture for the functions $f_{b,c,r}$

Let $m\ge 8$ be an even integer, let $b,c\in\mathbb{F}_{2^m}$, and let $r$ satisfy $r\le m/2$ and $\gcd(r,m)=1$. Let $f_{b,c,r}$ denote the function defined in Theorem. Non-APN conjecture. For every such choice of $b,c,r$, the function $f_{b,c,r}$ is not APN. The preceding computer checks establish this claim for $8\le m\le 22$ under the stated conditions, while the general case remains open.