Flag-transitive hyperplane complements in classical generalized quadrangles

Let H be a geometric hyperplane of a classical finite generalized quadrangle Q and let C = Q \ H be its complement in Q, viewed as a point-line geometry. We shall prove that C admits a flag-transitive automorphism group if and only if H spans a hyperplane of the projective space in which Q is naturally embedded (but with Q viewed as Q(4, q) when Q = W(q), q even). Furthermore, if Q is the dual of H(4, q²) and H, C are as above, then C is flag-transitive if and only if H = p^⊥ for some point p of Q.