Uniqueness of the projective plane underlying a partial geometry from a maximal arc

Let $G$ be a partial geometry with parameters $(s,t,\alpha)$ as in the paper, for integers $d\ge 2$ and $d'\ge 2$, and let $q=dd'$. Suppose that $G$ arises from a maximal arc of degree $d$ in a projective plane of order $q$ via the paper's Construction. Uniqueness conjecture. Either $G$ does not arise from such a maximal arc, or $G$ is obtainable from exactly one projective plane of order $q$ via that Construction. This conjecture formalizes the observed uniqueness of the projective plane in the known examples; its general status is not resolved in the supplied text.