Kantor's flag-transitive generalized quadrangle conjecture

Let $\mathcal{S}$ be a finite flag-transitive generalized quadrangle. A generalized quadrangle is classical when it is one of the standard generalized quadrangles arising from a classical polar space; two generalized quadrangles are considered equivalent up to duality when points and lines may be interchanged. Kantor's conjecture. If $\mathcal{S}$ is not a classical generalized quadrangle, then, up to duality, $\mathcal{S}$ is the unique generalized quadrangle of order $(3,5)$ or the generalized quadrangle of order $(15,17)$ arising from the Lunelli--Sce hyperoval. This is presented as part of the outstanding classification problem for flag-transitive generalized polygons; the supplied text gives no resolution.