Margulis's measure-rigidity conjecture for the diagonal action

Let $G={\rm SL}(n,\mathbb{R})$, $\Gamma={\rm SL}(n,\mathbb{Z})$, and $X=G/\Gamma$ for $n\geq 3$. Let $A<G$ be the group of positive diagonal matrices with determinant one. A Borel probability measure on $X$ is algebraic if there exists a closed subgroup $L<G$ such that $A<L$ and the measure is the unique $L$-invariant Borel probability measure on a single, closed $L$-orbit on $X$. Margulis's measure-rigidity conjecture. Every $A$-invariant and ergodic Borel probability measure $\mu$ on $X$ is algebraic. This conjecture would classify the invariant ergodic measures for the diagonal action, a rigidity property closely connected with the study of Littlewood's conjecture. The conjecture is explicitly stated as still open in the source.