The uniform-exclude-distribution maximality conjecture for APN graphs

Let $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ be an almost perfect nonlinear (APN) function. Let $\mathcal{G}_F$ be its graph, and let $d_{\mathcal{G}_F}$ be its exclude-distribution function on the partition $\mathcal{Q}(\mathbb{F}_2^n,F)$.

Uniform-exclude-distribution conjecture. If $d_{\mathcal{G}_F}$ is uniform on $\mathcal{Q}(\mathbb{F}_2^n,F)$, then $\mathcal{G}_F$ is maximal.

Uniformity on this partition imposes a stronger condition on a graph that is non-maximal. The conjecture is proposed as a direction for future work, and no resolution is given in the supplied text.