Let G be a compact Lie subgroup of $SU(n+1)$, and let $L=G\cdot p$ be a Lagrangian submanifold in $\mathbb{CP}^n$. If $\Omega$ denotes the $G^{\mathbb C}$ orbit through $p$, I know that $L$ is a deformation retract of $\Omega$ (since the gradient flow of the norm squared of the moment map defines a deformation retract of the appropriate piece of the manifold onto the zero level). Let $\pi$ denote the retraction from $\Omega$ to $L$. Is it already known when $\pi$ is a Riemannian submersion?