Every special set of the Hermitian surface $\mathsf{H}(3,q^2)$ is classical

Special sets of the Hermitian surface $\mathsf{H}(3,q^2)$, $q$ odd, were introduced by Shult and Thas (1995) in order to construct new finite generalised quadrangles, yet only one example is known to exist and it gives rise to a classical generalised quadrangle.

We show that there can be no other special sets of the Hermitian surface.