Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group containing $G$. We can construct now $G_3:=1+\operatorname{rad}(KG_2)$ and so on. By this way can construct a sequence of p-groups with growing quantities.
My question is, whether the exponent of these groups is bounded. As theses sequences are growing the bounding is equivalent to the fact that these exponents are getting constant after finite many steps.
Conjecture: The wreath product $\underbrace{C_p\wr ... \wr C_p}_{n-times}$ is involved in $G_n$.
