Classify gain functions on affine planes yielding generalized quadrangles

Determine all gain functions on an affine plane over a field that yield a generalized quadrangle through Construction $\mathfrak{M}$, and determine whether two distinct such gain functions on the same affine plane produce isomorphic generalized quadrangles.

Background

Theorem 7 gives one explicit gain function on the incidence graph of an affine plane over a field and proves that the associated Construction $\mathfrak{M}$ produces a generalized quadrangle.

The paper leaves unresolved the classification of every gain function on an affine plane with this property. It also asks whether distinct qualifying gain functions can nevertheless lead to isomorphic generalized quadrangles, making both the gain-function classification and the resulting generalized-quadrangle isomorphism problem explicit open questions.