The capacity conjecture for stably non-displaceable subsets

Let $(M,\omega)$ be a closed symplectic manifold, let $n$ be a positive integer, and let $X$ be a compact subset that is $n$-stably non-displaceable, meaning that $X\times T^n$ is not displaceable in $M\times T^\ast T^n$. For $e=(e_1,\ldots,e_n)\in\mathbb{Z}^n$ and $R=(R_1,\ldots,R_n)\in(\mathbb{R}_{>0})^n$, define the relative symplectic capacity by

$$C(M,X,R;e)=C_{BPS}(M\times I_R^n\times T^n,X\times T^n;(0_M,e)).$$

The capacity conjecture. The equality

$$C(M,X,R;e)=\sum_{i=1}^n R_i\cdot|e_i|$$

holds for every $e\in\mathbb{Z}^n$ and $R\in(\mathbb{R}_{>0})^n$. This predicts an explicit value for the Biran–Polterovich–Salamon capacity associated with any such compact subset.