Rigidity conjecture for holomorphic parabolic geometries

Let $G$ be a complex simple Lie group and let $P\subset G$ be a maximal parabolic subgroup. Assume that $G/P$ is not a compact Hermitian symmetric space, or, if $G/P$ is a compact Hermitian symmetric space, that $G$ is a proper subgroup of the identity component of the biholomorphism group of $G/P$. A holomorphic parabolic geometry modelled on $G/P$ is a holomorphic parabolic geometry whose model is $G/P$; a standard flat parabolic geometry is the flat model geometry on $G/P$. Rigidity conjecture. Up to isomorphism, the only holomorphic parabolic geometry modelled on $G/P$ on any compact Kähler manifold is the standard flat parabolic geometry on $G/P$. This conjecture asserts rigidity for holomorphic parabolic geometries on compact Kähler manifolds beyond the compact Hermitian symmetric cases, and the source presents it as the motivation for the rigidity results proved in the paper. Its resolution is not specified in the supplied text.