Kochetov's PI conjecture for smash products of hyperalgebras

Let $H$ be a hyperalgebra over a perfect field $\mathbb{F}$. If $\operatorname{char}\mathbb{F}>0$, assume also that $H$ is reduced. Let $G$ be a group acting on $H$ by bialgebra automorphisms.

Kochetov's conjecture. The smash product $H\#\mathbb{F}G$ is PI if and only if

$$\begin{array}{ll} \text{(i)} & H\text{ is commutative};\\ \text{(ii)} & \text{there exist normal subgroups }G_0\subset G_1\subset G\text{ such that }G/G_1\text{ is finite},\\ & G_1/G_0\text{ is abelian, and }G_0\text{ is a finite }p\text{-group if }\operatorname{char}\mathbb{F}=p>0\\ & \text{and trivial if }\operatorname{char}\mathbb{F}=0;\\ \text{(iii)} & G_1\text{ acts trivially on }H. \end{array}$$

This conjecture proposes a complete PI criterion for smash products of connected Hopf algebras by combining commutativity of the hyperalgebra with a finite-by-abelian structure condition on the acting group. The statement extends known criteria for enveloping algebras, restricted enveloping algebras, and group algebras, but the general case remains open.