The strengthened point-modification conjecture for APN functions

Let $F:{\mathbb F}_{2^n}\to{\mathbb F}_{2^n}$ be an APN function. Fix $x_0\in{\mathbb F}_{2^n}$ and let $F'$ be its $(x_0,\epsilon)$-modification, meaning that $F'(x_0)=F(x_0)+\epsilon$, $F'(x)=F(x)$ for $x\ne x_0$, and $\epsilon\ne0$. A function is $x_0$-APN when, for every $a\ne0$, the equation $F'(x+a)+F'(x)=F'(x_0+a)+F'(x_0)$ has only two solutions. Strengthened point-modification conjecture. An $(x_0,\epsilon)$-modification of an APN function with $\epsilon\ne0$ is not $x_0$-APN. By the theorem immediately preceding this conjecture, an $x_0$-APN modification would also be APN, so this strengthens the earlier point-modification conjecture. Its status is not resolved in the supplied text.