  We consider the finite $W$-algebra $U(\g,e)$ associated to a nilpotent
element $e \in \g$ in a simple complex Lie algebra $\g$ of exceptional type.
Using presentations obtained through an algorithm based on the PBW-theorem, we
verify a conjecture of Premet, that $U(\g,e)$ always has a 1-dimensional
representation, when $\g$ is of type $G_2$, $F_4$, $E_6$ or $E_7$. Thanks to a
theorem of Premet, this allows one to deduce the existence of minimal dimension
representations of reduced enveloping algebras of modular Lie algebras of the
above types. In addition, a theorem of Losev allows us to deduce that there
exists a completely prime primitive ideal in $U(\g)$ whose associated variety
is the coadjoint orbit corresponding to $e$.
