The irreducibility conjecture for fibers of conical Hamiltonian varieties

Let $G$ be a connected reductive group and let $X$ be a conical irreducible Hamiltonian $G$-variety. Property (Irr) means that every fiber of the quotient morphism

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

is irreducible.

Irreducibility conjecture. Every conical irreducible Hamiltonian $G$-variety $X$ satisfies (Irr).

This conjecture concerns the geometry of invariant moment maps. The paper gives a counterexample to (Irr) for a nonconical Hamiltonian variety, but no counterexample is known in the conical irreducible case; the conjecture is presented as open.