The nilpotent-orbit chart conjecture for generalized punctual Hilbert schemes

Let $\mathfrak{g}$ be a semisimple Lie algebra, and suppose a smooth version of $\operatorname{Hilb}(\mathfrak{g})$ exists. For classical $\mathfrak{g}$, let $m$ be the dimension of its standard representation and let $W$ be the relevant Weyl-group action on $\mathbb{C}[x,y]$. Nilpotent-orbit chart conjecture. The smooth version of $\operatorname{Hilb}(\mathfrak{g})$ is covered by charts parametrized by nilpotent orbits and all these charts are necessary. In particular, for classical $\mathfrak{g}$, the modified version of $\operatorname{Hilb}_0(\mathfrak{g})$ is isomorphic to the space of ideals of $\mathbb{C}[x,y]$ which are of codimension $m$, $W$-invariant, supported at $0$, and which lie in a chart associated to a partition of type $\mathfrak{g}$. The paper motivates these charts using the parametrization of nilpotent orbits, but does not prove the conjectural description.