Determine the isomorphism type of the affine-plane gain-function generalized quadrangle

Determine whether, for every finite field of order q, the generalized quadrangle GQ(q+1,q-1) constructed from the specified gain function on an affine plane is isomorphic to the dual of the Payne generalized quadrangle P(W(q),x).

Background

The paper constructs a generalized quadrangle from an affine plane over an arbitrary field by assigning an additive gain function to the incidence graph. For finite fields of order q, the resulting generalized quadrangle has parameters GQ(q+1,q-1).

The author reports computer verification of an isomorphism with the dual of the Payne generalized quadrangle P(W(q),x) for q up to 16, but explicitly states this as a conjecture for all finite field orders. Thus, the unresolved issue is whether the observed identification holds universally.

References

When $\mathbb{F}$ is finite, say order $q$, we checked by computer that the $GQ(q+1,q-1)$ constructed in Theorem 7 is isomorphic to the dual of the Payne GQ $P(W(q),x)$. This has been confirmed for values of $q$ up to $16$, and we conjecture that it is always true.

— Incidence Gain Graphs and Generalized Quadrangles  (2502.01805 - McCulloch, 3 Feb 2025) in Section “Affine Plane Examples and Further Inquiry”