Nonexistence conjecture for doubly regular generalized quadrangles

Let $\mathcal{Q}$ be a generalized quadrangle of order $s>1$, and suppose a group of automorphisms acts regularly on both the point set and the line set of $\mathcal{Q}$. Nonexistence conjecture. No such generalized quadrangle exists. This conjecture is motivated by the impossibility of a polarity for a thick generalized quadrangle with such a regular automorphism group and by severe restrictions on the order of $\mathcal{Q}$ and the acting group. The source provides numerical evidence but does not establish the conjecture in general.