A conjectural chain model for positive $S^1$-equivariant symplectic homology of star-shaped toric domains in ${\mathbb{C}}^2$

For any star-shaped toric domain in ${\mathbb{C}}^2$, we define a filtered chain complex which conjecturally computes positive $S^1$-equivariant symplectic homology of the domain. Assuming this conjecture, we show that the limit ${\lim }_{k\to \mathrm{\infty }}{c}_k^{\mathrm{G}\mathrm{H}}(X)\mathrm{/}k$ exists for any star-shaped toric domain $X\subset {\mathbb{C}}^2$, where $c_k^{\mathrm{G}\mathrm{H}}$ denotes the $k$-th Gutt-Hutchings capacity.