Absence of collinear quadruples in minimizing generalized permutations

Prove or disprove that every generalized permutation in a finite affine plane minimizing the number of collinear triples contains no collinear quadruples.

Background

For permutations attaining the minimum (q−1)/2 collinear triples in AG(q,2), the paper derives that no four graph points are collinear, because otherwise the relevant Bonferroni inequality would be strict and contradict the known lower bound. The unresolved question is whether this structural property extends from Desarguesian affine planes to all finite affine planes and their minimizing generalized permutations.