Classification of power-of-two-sized k-covers

Prove or disprove that every k-cover S⊆F_2^n with |S|=2^{n/2} is affinely equivalent to the graph of an almost bent function F:F_2^{n/2}→F_2^{n/2}.

Background

The paper proves that if a Sidon set S of affine dimension n has size 2^{n/2}, then γ_S is bent if and only if S is a k-cover. Graphs of almost bent functions provide examples of such k-covers, but it is unresolved whether all k-covers of this size arise from almost bent functions up to affine equivalence.

The authors formulate the unresolved classification as a conjecture asserting that every such k-cover is affinely equivalent to the graph of an almost bent function.