Symplectic duality cohomology and fixed-point coordinate-ring conjecture

Let $X\to X_0$ and $X^!\to X_0^!$ be symplectic dual conical symplectic resolutions, and let a torus $\mathbb{T}$ act on $X_0^!$. Denote by $H^*(X)$ the cohomology of $X$ and by the coordinate ring of $\mathbb{T}$-fixed points in $X_0^!$ the corresponding graded algebra. Symplectic duality conjecture. The cohomology $H^*(X)$ and the coordinate ring of $\mathbb{T}$-fixed points in $X_0^!$ are isomorphic as graded algebras, and vice versa. This predicts a relation between the topology of one member of a symplectic-dual pair and the fixed-point coordinate ring of the other. In the cases numbered (1), (4), and (5), the conjecture has been proved by Hikita; the analogous assertion in the opposite direction is included in the statement.