  Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in
$G$, let $O$ be an open subset of $X$, and let $F = \{g_t: t\ge 0\}$ be a
one-parameter subsemigroup of $G$. Consider the set of points in $X$ whose
$F$-orbit misses $O$; it has measure zero if the flow is ergodic. It has been
conjectured that this set has Hausdorff dimension strictly smaller than the
dimension of $X$. This conjecture is proved when $X$ is compact or when $G$ is
a simple Lie group of real rank $1$, or, most recently, for certain special
flows on the space of lattices. In this paper we prove this conjecture for
arbitrary $\operatorname{Ad}$-diagonalizable flows on irreducible quotients of
semisimple Lie groups. The proof uses exponential mixing of the flow together
with the method of integral inequalities for height functions on $G/\Gamma$. We
also derive an application to jointly Dirichlet-Improvable systems of linear
forms.
