Identification with the dual of the Payne generalized quadrangle

Prove that, for every finite field of order \(q\), the generalized quadrangle \(GQ(q+1,q-1)\) constructed in Theorem 7 is isomorphic to the dual of the Payne generalized quadrangle \(P(W(q),x)\).

Background

For finite fields of order $q$, the construction in Theorem 7 produces a generalized quadrangle denoted $GQ(q+1,q-1)$. The author reports computational verification that this generalized quadrangle is isomorphic to the dual of the Payne generalized quadrangle $P(W(q),x)$ for values of $q$ up to 16.

The reported computations motivate the explicit conjecture that the isomorphism holds for every finite field order, but the paper does not prove this assertion. The unresolved problem is therefore to establish the claimed identification uniformly for all finite $q$.