Margulis's algebraicity conjecture for invariant measures

Let $G$, $\Gamma$, and $H$ be as in Margulis's orbit-closure conjecture, and let $\mu$ be an $H$-invariant, $H$-ergodic measure on $\Gamma\backslash G$. Suppose that for every $x\in\operatorname{supp}\mu$, alternative (2) of Margulis's orbit-closure conjecture does not hold. Margulis's measure-rigidity conjecture. Then $\mu$ is algebraic, meaning that it is the invariant measure on the closed orbit of some subgroup $L<G$ containing $H$.

This is the measure-theoretic counterpart of Margulis's orbit-closure conjecture and extends the classification paradigm of Ratner's theorem. The source presents it as open.