  We prove a number field analogue of W. M. Schmidt's conjecture on the
intersection of weighted badly approximable vectors and use this to prove an
instance of a conjecture of An, Guan and Kleinbock. Namely, let $G :=
SL_2(\mathbb{R}) \times \dots \times SL_2(\mathbb{R}) $ and $\Gamma$ be a
lattice in $G$. We show that the set of points on $G/\Gamma$ whose forward
orbits under a one parameter Ad-semisimple subsemigroup of $G$ are bounded,
form a hyperplane absolute winning set.
