Isomorphism with the dual Payne generalized quadrangle

Determine whether, for every finite field of order q, the generalized quadrangle GQ(q+1,q-1) constructed from the gain function in Theorem 7 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 a field by using the additive action of the field and an explicitly defined gain function on the affine-plane incidence graph. For a finite field of order q, the resulting generalized quadrangle has parameters GQ(q+1,q-1).

The author reports computational confirmation of an isomorphism with the dual of the Payne generalized quadrangle P(W(q),x) for values of q up to 16, but does not establish the claim for all finite fields. The universal isomorphism assertion is therefore presented as a conjecture.