Alexander Olshanskii has pointed out to me that Hanna Neumann [10] noted that varieties of Burnside groups of appropriate prime exponents give negative answers to this, and that another sort of counterexample is given in [11].

[10] Hanna Neumann, *Varieties of groups,*
Springer-Verlag 1967 x+192 pp.
MR0215899

[11] A. Yu. Olshanskii,
*Varieties in which all finite groups are abelian,*
Matem. Sbornik, **126** (1985) N.1, 59-82.
MR0773429