The generalized GIT quotient conjecture for generalized punctual Hilbert schemes

Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $\operatorname{Hilb}(\mathfrak{g})$ denote its generalized Hilbert scheme. Generalized GIT quotient conjecture. There is a generalized GIT quotient procedure identifying infinitesimally close points in $\operatorname{Hilb}(\mathfrak{g})$, giving a modified $\mathfrak{g}$-Hilbert scheme which is Hausdorff, and even smooth. This is proposed as a replacement for the ordinary GIT quotient, which does not apply to the nonclosed locus considered in the paper; no such procedure is established there.