Let $(M,\omega)$ be a symplectic manifold.

I'm trying to show that a compact subset $K\subset M$ is $1$-superheavy [1, Definition 1.3] where $1=PD([M])$ is the unit in $QH^0(M)$.

My question is: How do symplectic invariants of $K^c=M\setminus K$ influence this property? I'm thinking about invariants like $c_1(K^c)$, $QH^*(K^c)$ and $SH^*(K^c)$.

An interesting special case would be where $c_1(K^c)=0$ or $SH^*(K^c)=0$.

I have the feeling that the answer goes through properties of symplectic quasi-states (something like: as $SH^*(K^c)=0$ the quasi-state is only affected by the behaviour on $K$), but can't see how.

[1] Rigid subsets of symplectic manifolds, Entov and Polterovich