The nilpotent-model irreducibility conjecture for Hamiltonian fibers

Let $G$ be a connected reductive group, let $H$ be a subgroup, let $\eta$ be a nilpotent element, and let $V$ be an $H$-variety. Write $X=M_G(H,\eta,V)$ for the associated Hamiltonian model variety. Property (Irr) means that every fiber of the quotient morphism

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

is irreducible.

Nilpotent-model irreducibility conjecture. If $X=M_G(H,\eta,V)$, where $\eta$ is nilpotent, then $X$ satisfies (Irr).

The paper states this as a weaker version of the conjecture for all conical irreducible Hamiltonian varieties. It notes that this conjecture would imply normality of every fiber of $\psi_{G,X}$ and suggests a reduction to case-by-case verification; it remains open.