Alexander Olshanskii has pointed out to me that Hanna Neumann  noted that varieties of Burnside groups of appropriate prime exponents give negative answers to this, and that another sort of counterexample is given in .
 Hanna Neumann, Varieties of groups, Springer-Verlag 1967 x+192 pp. MR0215899
 A. Yu. Olshanskii, Varieties in which all finite groups are abelian, Matem. Sbornik, 126 (1985) N.1, 59-82. MR0773429