Regev's and Amitsur's conjectures for codimensions of generalized polynomial identities

Let $A$ be a finite-dimensional associative algebra over a field of characteristic 0. Then there exist $C\in\mathbb Q_+$ and $t\in\mathbb Z_+$ such that $\mathrm{gc}_n(A)\sim Cn^td^n$ as $n\to\infty$, where $d=\mathrm{PI}\exp(A)$. In particular, Amitsur's and Regev's conjectures hold for the codimensions $\mathrm{gc}_n(A)$ of generalized polynomial identities.