Bahturin–Zaicev codimension-growth inequality for finite group gradings

Let $W$ be an associative PI-algebra over a field $F$ of characteristic zero, graded by a finite group $G$. Let $\exp(W)$ and $\exp(W_e)$ denote the codimension-growth exponents of $W$ and its identity component $W_e$, respectively.

Bahturin–Zaicev conjecture. One has

$$\exp(W) \leq |G|^{2}\exp(W_e).$$

This inequality was conjectured for general PI-algebras and was known in the affine case. The paper proves it in general.