The APN distance-one conjecture

Let $n\geq 3$, and let $F,G\colon\mathbb{F}_2^n\to\mathbb{F}_2^n$ be APN functions. Define their Hamming distance by

$$d(F,G)=\left|\left\{x\in\mathbb{F}_2^n:F(x)\neq G(x)\right\}\right|.$$

APN distance-one conjecture. If $F\neq G$, then $d(F,G)>1$. It is unknown whether two APN functions can have Hamming distance one; the conjecture is known for several classes, including known APN power functions except the Dobbertin function and APN plateaued functions.