Equivalence of the third family of quadratic APN permutations

Determine whether the third listed family of quadratic APN permutations is inequivalent to the Gold-function family and the BudaghyanCarletLeander family.

Background

The paper surveys three known families of crooked functions, all of which are quadratic APN permutations. The first family consists of Gold functions, while the second is the BudaghyanCarletLeander family; the paper states that these two families have been proved EA-inequivalent.

The third family, constructed by Li and Kaleyski, is known to contain distinct EA-inequivalent functions for different parameter choices, but its equivalence relation with the first two families has not been established. Resolving this conjecture would clarify the diversity of crooked functions and the potential inequivalence of the resulting line-parallelisms.