Margulis's conjecture on Hilbert modular lattices

Let $G$ be the product of $n\geq 2$ copies of $\operatorname{SL}(2,\mathbb{R})$. Let $U_1$ and $U_2$ be the products of, respectively, upper- and lower-unipotent one-parameter subgroups in the factors of $G$. Let $\Gamma<G$ be discrete, assume that for $i=1,2$, $\Gamma\cap U_i$ is a lattice in $U_i$, and assume that for every proper connected normal subgroup $N<G$, the intersection $\Gamma\cap N\cap U_i$ is trivial. Margulis's conjecture. Then $\Gamma$ is commensurable with a Hilbert modular lattice, up to conjugation in $\operatorname{GL}(2,\mathbb{R})\times\cdots\times\operatorname{GL}(2,\mathbb{R})$. This is a rigidity conjecture for discrete subgroups of products of rank-one groups. The source mentions an application due to Hee Oh but supplies no resolution of the conjecture.