Group graded PI-algebras and their codimension growth

Let W be an associative PI — algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W) and exp(We) denote the codimension growth of W and of the identity component We, respectively. The following inequality had been conjectured by Bahturin and Zaicev: exp(W) ≤ |G|2 exp(We). The inequality is known in case the algebra W is affine (i.e., finitely generated). Here we prove the conjecture in general.