On the number of quadratic orthomorphisms that produce maximally nonassociative quasigroups

Let q be an odd prime power and suppose that a,b in F_q are such that ab and (1-a)(1-b) are nonzero squares. Let Q_a,b = (F_q,*) be the quasigroup in which the operation is defined by u*v=u+a(v-u) if v-u is a square, and u*v=u+b(v-u) is v-u is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies x*(y*z) = (x*y)*z ⇔ x=y=z. Denote by σ(q) the number of (a,b) for which Q_a,b is maximally nonassociative. We show that there exist constants α ≈ 0.02908 and β ≈ 0.01259 such that if q= 1 mod 4, then lim σ(q)/q² = α, and if q = 3 mod 4, then lim σ(q)/q² = β.