  We prove a modified version for a conjecture of Weiss from 2004. Let $G$ be a
semisimple real algebraic group defined over $\mathbb{Q}$, $\Gamma$ be an
arithmetic subgroup of $G$. A trajectory in $G/\Gamma$ is divergent if
eventually it leaves every compact subset, and is obvious divergent if there is
a finite collection of algebraic data which cause the divergence. Let $A$ be a
diagonalizable subgroup of $G$ of positive dimension. We show that if the
projection of $A$ to any $\mathbb{Q}$-factor of $G$ is of small enough
dimension (relatively to the $\mathbb{Q}$-rank of the $\mathbb{Q}$-factor),
then there are non-obvious divergent trajectories for the action of $A$ on
$G/\Gamma$.
