Second largest maximal cliques in small Paley graphs of square order

There is a conjecture that the second largest maximal cliques in Paley graphs of square order $P(q^2)$ have size $\frac{q+\epsilon }{2}$, where $q\equiv \epsilon (\mathrm{m}\mathrm{o}\mathrm{d}4)$, and split into two orbits under the full group of automorphisms whenever $q\ge 25$ (a symmetric description for these two orbits is known). However, some extra second largest maximal cliques (of this size) exist in $P(q^2)$ whenever $q\in \{9,11,13,17,19,23\}$. In this paper we analyse the algebraic and geometric structure of the extra cliques.