  Consider the W-algebra $W$ attached to the smallest nilpotent orbit in a
simple Lie algebra $\frak g$ over an algebraically closed field of
characteristic 0. We show that if an analogue of the Gelfand-Kirillov
conjecture holds for such a W-algebra then it holds for the universal
enveloping algebra $\mathrm U(\frak g)$. This together with a result of A.
Premet implies that the analogue of the Gelfand-Kirillov conjecture fails for
some $W$-algebras attached to some nilpotent orbits in Lie algebras of types
$B_n~(n\ge 3)$, $D_n~(n\ge 4)$, $E_6, E_7, E_8$, $F_4$.
