Bahturin's Amitsur-type conjecture for Hopf-module codimensions

Let $F$ be a field of characteristic $0$, let $H$ be a Hopf algebra over $F$, and let $A$ be a finite-dimensional associative $H$-module algebra, with codimensions

$$c_n^H(A):=\dim\left(\frac{P_n^H}{P_n^H\cap\operatorname{Id}^H(A)}\right),$$

where $P_n^H$ is the space of multilinear $H$-polynomials and $\operatorname{Id}^H(A)$ is the ideal of polynomial $H$-identities of $A$. Bahturin's Amitsur-type conjecture. There exists an integer

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

This is the Hopf-module analogue of Amitsur's conjecture on the asymptotic behaviour of codimensions of ordinary polynomial identities. The statement concerns the existence and integrality of the limit for finite-dimensional associative $H$-module algebras over characteristic-zero fields.