Uniqueness of the degree-three Kantor ovoid family in characteristic two

Determine whether, for every triple of degree-three polynomials f₁, f₂, f₃ over 𝔽_q having the form displayed in equation (16), the ovoid O₇(f₁,f₂,f₃) is equivalent to the Kantor ovoid described in Section 3.3.

Background

For characteristic p = 2 and q ≡ 2 (mod 3), the authors classify the possible degree-three polynomial triples arising when the associated hypersurface splits into two quadrics. The resulting family is given explicitly in equation (16).

The paper leaves unresolved whether every ovoid in this characteristic-two family is equivalent to the Kantor ovoid, so the classification is only up to the remaining equivalence question.