  Let $G$ be connected reductive algebraic group defined over an algebraically
closed field of characteristic $p > 0$ and suppose that $p$ is a good prime for
the root system of $G$, the derived subgroup of $G$ is simply connected and the
Lie algebra $\mathfrak{g} = \operatorname{Lie}(G)$ admits a non-degenerate
Ad$(G)$-invariant symmetric bilinear form. Given a linear function $\chi$ on
$\mathfrak{g}$ we denote by $U_\chi(\mathfrak{g})$ the reduced enveloping
algebra of $\mathfrak{g}$ associated with $\chi$. By the Kac-Weisfeiler
conjecture (now a theorem), any irreducible $U_\chi(\mathfrak{g})$-module has
dimension divisible by $p^{d(\chi)}$ where $2d(\chi)$ is the dimension of the
coadjoint $G$-orbit containing $\chi$. In this paper we give a positive answer
to the natural question raised in the 1990s by Kac, Humphreys and the
first-named author and show that any algebra $U_\chi(\mathfrak{g})$ admits a
module of dimension $p^{d(\chi)}$.
