The AB-graph conjecture for maximal k-covers

Let $S\subseteq \mathbb{F}_2^n$ be a $k$-cover, and suppose that

$$|S|=2^{n/2}.$$

An AB function is a function $F\colon\mathbb{F}_2^{n/2}\to\mathbb{F}_2^{n/2}$ whose graph is denoted by $\mathcal{G}_F$; two sets are affinely equivalent when they are related by an affine transformation. AB-graph conjecture. If $S\subseteq\mathbb{F}_2^n$ is a $k$-cover with $|S|=2^{n/2}$, then there exists an AB function $F\colon\mathbb{F}_2^{n/2}\to\mathbb{F}_2^{n/2}$ such that $\mathcal{G}_F$ is affinely equivalent to $S$. The theorem preceding the conjecture shows that every such $k$-cover gives a bent function, while graphs of AB functions provide examples; the conjecture asserts that these are the only examples up to affine equivalence.