Hamiltonian stability conjecture for compact semisimple Lagrangian orbits in complex projective space

Let $G$ be a compact (semi)simple subgroup of $\SU{N}$ for some $N$, and let $\mathcal O$ be a Lagrangian orbit of $G$ in $\CP^{N-1}$. Hamiltonian stability conjecture. If $G$ admits such a Lagrangian orbit, then $\mathcal O$ is Hamiltonian stable. The conjecture proposes a broad classification principle for minimal Hamiltonian stable Lagrangian submanifolds arising as homogeneous orbits; examples and stability results are known in several special cases, but the general assertion remains open.