Identities of graded algebras and codimension growth

Let $A=\oplus_{g\in G}A_g$ be a $G$-graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component $A_e$ to that of the whole of $A$, in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where $A$ is finite dimensional and $A_e$ has polynomial growth.