The axis-value conjecture for power APN functions

Suppose $n$ is odd, and let $F \colon \mathbb{F}_{2^n}\to\mathbb{F}_{2^n}$ be the power function

$$F(x)=x^d.$$

Assume that $F$ is almost perfect nonlinear (APN). Let $d_{\mathcal{G}_F}$ denote the exclude-distribution function of the graph of $F$.

Axis-value conjecture. For every $a\in\mathbb{F}_2^n$ and every nonzero $b\in\mathbb{F}_2^n$,

$$d_{\mathcal{G}_F}(a,0)=d_{\mathcal{G}_F}(0,b)=\frac{2^n-2}{6}.$$

The statement is presented as a conjectural pattern suggested by computer calculations for power functions. Its general validity is not established in the supplied text.