Uniqueness of the degree-three Kantor ovoid family for odd characteristic

Determine whether, for every triple of degree-three polynomials f₁, f₂, f₃ over 𝔽_q having the form displayed in equation (14), the ovoid O₇(f₁,f₂,f₃) is equivalent to the Kantor ovoid described in Section 3.3.

Background

The paper analyzes ovoids of the hyperbolic quadric Q⁺(7,q) parameterized by triples of polynomials f₁,f₂,f₃ of degree at most three. In the case q ≡ 2 (mod 3), with characteristic p > 3, the authors show that any degree-three ovoid arising from the two-quadrics factorization case must be described by the polynomial family in equation (14).

The Kantor ovoid for q ≡ 2 (mod 3) is known to have degree three and to belong to this family. The unresolved issue is whether the entire family in equation (14), after allowing the relevant equivalences, consists only of the Kantor ovoid.