CCZ invariance of distance to affine functions

Determine whether the parameter A_f, the minimum Hamming distance from an (n,m)-function f:GF(2)^n\to GF(2)^m to the affine functions, is invariant under CCZ-equivalence.

Background

The paper notes that several properties of vectorial Boolean functions, including differential uniformity and vectorial nonlinearity, are invariant under CCZ-equivalence. It introduces A_f as the distance from a vectorial Boolean function to the affine maps and gives a Walsh-transform formula for computing this parameter.

The authors explicitly leave unresolved whether A_f is preserved when the graph of one function is mapped to the graph of another by an invertible affine transformation of the product space GF(2)^n\times GF(2)^m.