The heavy–SH-heavy equivalence conjecture

Let $(M,\omega)$ be a closed symplectic manifold, and let $K\subset M$ be compact. The subset $K$ is heavy if it satisfies the heavy-set condition defined using the relevant symplectic quasi-state, and it is SH-heavy if it satisfies the corresponding symplectic-homology condition. Heavy–SH-heavy equivalence conjecture. A compact subset of a closed symplectic manifold is heavy if and only if it is SH-heavy. This conjecture proposes an equivalence between two notions of symplectic rigidity; the paper raises the connection between heavy and SH-heavy sets as an interesting question, without supplying a resolution.