Kantor's classification conjecture for flag-transitive generalized quadrangles

A generalized quadrangle is an incidence geometry whose incidence graph has diameter $4$ and girth $8$. It is flag-transitive if a group of collineations acts transitively on incident point-line pairs; two generalized quadrangles are identified up to duality when points and lines may be interchanged.

Kantor's conjecture. If $\mathcal{Q}$ is a finite flag-transitive generalized quadrangle and $\mathcal{Q}$ is not a classical generalized quadrangle, then, up to duality, $\mathcal{Q}$ is the unique generalized quadrangle of order $(3,5)$ or the generalized quadrangle of order $(15,17)$ arising from the Lunelli–Sce hyperoval.

This conjecture concerns the classification of finite flag-transitive generalized quadrangles. The two asserted non-classical examples are the generalized quadrangle of order $(3,5)$ and the example of order $(15,17)$ arising from the Lunelli–Sce hyperoval; the classification question is stated as an outstanding open problem.