On uniqueness of submaximally symmetric parabolic geometries

Among the (regular, normal) parabolic geometries of type $(G,P)$, there is a locally unique maximally symmetric structure and it has the symmetry dimension $\mathrm{\text{dim}}(G)$. The symmetry gap problem concerns the determination of the next realizable (submaximal) symmetry dimension. When $G$ is a complex or split-real simple Lie group of rank at least three or when $(G,P)=(G_2,P_2)$, we establish a local uniqueness result for submaximally symmetric structures of type $(G,P)$.