Let $(M^{2n},\omega)$ be a compact symplectic manifold. We say that $M$ has the hard Lefscthetz property iff the homomorphisms of cohomology $$L^{k}:H^{n-k}(M)\longrightarrow H^{n+k}(M)$$ defined by setting $$ \alpha\mapsto \alpha\wedge\omega^{k} $$ are isomorphisms for $0<k<n$. A symplectic manifold satisfying the hard Lefschetz property is called a Lefschetz symplectic manifold. Suppose that $M$ is a Hamiltonian $S^{1}$-symplectic manifold, then for any closed subgroup $H\subseteq S^{1}$ the fixed point submanifold $M^{H}$ is also a symplectic submanifold with symplectic form $\omega|_{M^{H}}$. In the paper "Examples of non-kahler hamiltonian circle manifolds with the strong Lefschetz property", Yi Lin showed that there exist infinitely many Lefschetz symplectic 6-manifolds which admit non-Lefschetz fixed point submanifolds. My question is:
Consider a compact Lefschetz symplectic manifold with a Hamiltonian $S^{1}$-action. When is the fixed submanifold $M^{H}$ also a Lefschetz symplectic manifold for every closed subgroup $H\subseteq S^{1}$?