Kawasaki's capacity equality conjecture for stably non-displaceable subsets

Let $X$ be a stably non-displaceable compact subset of a closed symplectic manifold $(M,\omega)$. Kawasaki's conjecture. For any $R>0$ and $\ell\in\mathbb{Z}$, one has

$$C(M,X;R,0,\ell,-\infty)=R\lvert\ell\rvert.$$

This conjecture proposes an exact formula for the relative symplectic capacity associated with stably non-displaceable subsets; it generalizes the capacity equality established in the torus case, while its validity in the stated generality remains open.