The symplectic-singularities conjecture for untwisted nilpotent models

Let $G$ be a connected reductive group, let $H$ be a subgroup, let $\eta$ be nilpotent, and let $V$ be an $H$-variety. Set $X=M_G(H,\eta,V)$, and suppose that $X$ is untwisted. For a fiber $Y$ of

$$\widetilde{\psi}_{G,X}/\!/G,$$

say that $Y$ has symplectic singularities if there is a resolution of singularities $\widetilde{Y}\to Y$ such that the symplectic form on the smooth part of $Y$ extends to a regular form on $\widetilde{Y}$.

Symplectic-singularities conjecture. Every fiber $Y$ of $\widetilde{\psi}_{G,X}/\!/G$ has symplectic singularities.

This conjecture strengthens assertion 1 of Theorem 0.5 and concerns the Poisson structure on the fibers, whose open stratum is symplectic. It is stated as an open problem for untwisted nilpotent model varieties.