The moment-map fiber dimension conjecture

Let $G$ be a reductive group and let $X$ be an irreducible affine Hamiltonian $G$-variety. Let $\mu_{G,X}:X\to\mathfrak g$ be its moment map, let $m_G(X)$ and $\operatorname{def}_G(X)$ denote the invariants used in the paper, and let $G\eta$ be the $G$-orbit of $\eta\in\mathfrak g$.

Moment-map fiber dimension conjecture. One has

$$\dim\mu_{G,X}^{-1}(\eta)\leq \dim X-\frac{m_G(X)+\operatorname{def}_G(X)+\dim G\eta}{2}.$$

This conjecture proposes a uniform upper bound for dimensions of moment-map fibers in terms of the complexity-type invariant $m_G(X)$, the defect, and the dimension of the orbit of $\eta$. The paper presents it as an open problem.