Let $k$ be an algebraically closed field of characteristic zero, $V$ a finite dimensional $k$ vector space, $V^{\ast}$ the dual space, and $G$ an algebraic subgroup of $GL(V)$. Let $V_0$ be the points in $V$ whose stabilizer has no nontrivial subtorus (in other words, an extension of a finite group by a unipotent group) and define $V^{\ast}_0$ likewise.
Kac's conjecture $(\ast)$ states (citation below) that the number of $G$-orbits on $V_0$ and $V_0^{\ast}$ are equal.
Has this conjecture been proved? All I really need is that $V_0$ is nonempty if and only if $V^{\ast}_0$ is. I skimmed through the papers in Mathscinet citing Kac, but there are 148 of them, so it is easy to believe I missed it.
Kac, V.G., Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56, 57-92 (1980). ZBL0427.17001.
