Let a complex reductive group $G$ act on a $\mathbb{C}^{n}$ with finitely many orbits. Let $\mathcal{S}$ be the stratification of $\mathbb{C}^{n}$ according to these orbits. Let $(x,\xi) \in T_S^{*}\mathbb{C}^{n}$ be a point in the conormal to the orbit $S$. $(x,\xi)$ is called $\mathcal{S}$-regular if it isn't in the closure of $T^{*}_{S'}\mathbb{C}^{n}$ for any orbit $S' > S$.
Are there any known sufficient conditions for $(x,\xi)$ to be $\mathcal{S}$-regular?
I have one guess which seems to be true in the simple examples I'm coming up with, but I can't find a proof or a reference. The stabilizer $H = \mathrm{Stab}_G(x)$ acts on the fibre $(T_S^*\mathbb{C}^{n})_x$.
Conjecture: If $\mathrm{Stab}_H(x,\xi) = 1$ then $(x,\xi)$ is $\mathcal{S}$-regular
Is this known to be true, or is there an easy counterexample that I'm just missing?
