Non-trivial automorphism conjecture for APN functions and permutations

Let $F\colon\mathbb{F}_2^n\rightarrow\mathbb{F}_2^n$ be an APN function, and let $\mathsf{Aut}(F)$ denote its automorphism group. If $F$ is an APN permutation, let $\mathsf{Aut_{LE}}(F)$ denote its group of LE-automorphisms. Two functions are CCZ-equivalent when their graphs are related by an affine permutation of the product space. Non-trivial automorphism conjecture. For every APN function $F\colon\mathbb{F}_2^n\rightarrow\mathbb{F}_2^n$, we have

$$|\mathsf{Aut}(F)|>1.$$

Moreover, if $F$ is an APN permutation, there exists a CCZ-equivalent permutation $G$ such that

$$|\mathsf{Aut_{LE}}(G)|>1.$$

All known APN functions have non-trivial automorphism groups, and all known APN permutations have a CCZ-equivalent representative with a non-trivial LE-automorphism; the conjecture asserts that these observed properties hold in general.