Amitsur's conjecture on codimension growth of associative PI algebras

Let $A$ be an associative p.i. algebra, meaning an algebra satisfying a polynomial identity that does not hold in all associative algebras, over a field of characteristic $0$. Let $c_n(A)$ denote its $n$th codimension, and write

$$\operatorname{PIexp}(A):=\lim\limits_{n\to\infty}\sqrt[n]{c_n(A)}.$$

Amitsur's conjecture. The limit exists and belongs to $\mathbb Z_+$.

Amitsur's conjecture describes the asymptotic behavior of codimensions of associative PI algebras. It was proved in 1999 by A. Giambruno and M. V. Zaicev.