Margulis's almost-everywhere 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 $\mu$-almost every $x$, alternative (2) of Margulis's orbit-closure conjecture does not hold. Margulis's modified measure-rigidity conjecture. Then $\mu$ is algebraic.

The source proposes this as a less restrictive modification of Margulis's original measure conjecture, replacing a condition on every point of the support by an almost-everywhere condition. It remains open in the source.