On the codimension growth of G-graded algebras

Let W be an associative PI-affine algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W) and exp(W_e) denote the codimension growth of W and of the identity component W_e, respectively. We prove: exp(W) \leq |G|^2 exp(W_e). This inequality had been conjectured by Bahturin and Zaicev.