All varieties here are over $\Bbb C$. Let $G$ be a reductive algebraic group acting algebraically on affine $n$-space $\Bbb A^n$. Let $R$ be the coordinate ring of $\Bbb A^n$. Assume that the natural morphism $\pi\colon \Bbb A^n \to X=\operatorname{Spec}(R^G)$ is an almost geometric quotient.

Question: Are there "useful" (interpret as you will) conditions on $G$ which imply that $\pi$ is flat? The cases I care about are when $G$ is a closed subgroup of $(\Bbb G_m)^n$ acting via the induced action.