Quadraticity of crooked functions

Prove or disprove that every crooked function over a binary finite field is quadratic.

Background

Crooked functions are a class of functions over finite fields of characteristic two used to construct Preparata-like codes and, through the paper's coloring construction, line-parallelisms of binary projective spaces. The paper notes that all currently known crooked functions are quadratic, specifically quadratic APN permutations.

The unresolved conjecture concerns whether this observed quadratic structure is universal. Establishing it would classify the possible crooked functions and clarify whether genuinely nonquadratic examples can produce further Preparata-like codes and associated line-parallelisms.