The symplectic zero-fiber conjecture for generalized punctual Hilbert schemes

Let $\mathfrak{g}$ be a Lie algebra, let $\operatorname{Hilb}(\mathfrak{g})$ be its generalized Hilbert scheme, and let $\operatorname{Hilb}_0(\mathfrak{g})$ be the zero-fiber of the Chow map. Symplectic zero-fiber conjecture. The conjectural smooth version of the $\mathfrak{g}$-Hilbert scheme is symplectic and the zero-fiber is a Lagrangian subspace. For classical $\mathfrak{g}$, the analogous assertion is proved on the regular part; the general claim is based on the conjectural smooth modification and remains open.