# Is the symplectic quotient $\mu^{-1}(0)/G$ unique up to something?

Given a Hamiltonian action of a compact Lie group $$G$$ on a symplectic manifold $$(M,\omega)$$, we may choose a moment map $$\mu \colon M\to \mathfrak{g}^*$$ and obtain the symplectic reduction $$M/\!\!/G = \mu^{-1}(0)/G$$. This construction clearly depends on the choice of moment map. However, I wonder if it is still unique up to some sort of (very?) weak equivalence in the symplectic category?

When $$G$$ is a torus, any constant addition to a moment map is also a moment map. In a paper "Birational equivalence in the symplectic category (1989)" by Guillemin and Sternberg, authors showed that reduced spaces at regular levels are related by blowing up and down. I do not know the recent progress though. It might be helpful to read papers citing their paper.
The other extreme case is when $$G$$ is semisimple. In this case, the moment map, which is $$G$$-equivariant, is unique by the semisimplicity. Then the reduced space is unique and there is nothing to do.