Symplectic duality conjecture for Poisson homology and intersection cohomology

Let $\mathfrak{N}$ and $\mathfrak{N}^!$ be a symplectic dual pair of cones admitting projective symplectic resolutions $\mathfrak{M}$ and $\mathfrak{M}^!$. Let $\mathscr{N}$ be the quasi-universal deformation of $\mathfrak{N}$, and let $T^!$ be a maximal torus in the Hamiltonian automorphism group of $\mathfrak{M}^!$. Symplectic duality conjecture. There exist canonical isomorphisms of graded vector spaces

$$H\!P_0(\mathfrak{N})\cong I\!H^*(\mathfrak{N}^!) \qquad\text{and}\qquad H\!P_0(\mathscr{N})\cong I\!H^*_{T^!}(\mathfrak{N}^!),$$

where the second isomorphism is compatible with the module structure over $\mathbb{C}[\operatorname{H}^2(\mathfrak{M})]\cong \operatorname{H}^*(BT^!)$. The theorem preceding this conjecture proves the assertion for hypertoric varieties. A general definition of symplectic dual pairs was still in preparation, and the general assertion remains open in the supplied text.