Stabilization conjecture for cohomology rings of affine group schemes

Let $p$) be a prime number, and let $\mathbb{G}(-)$ be an affine group scheme over the $p$-adic integers $\mathbb{Z}_p$. For each $n\geq 1$, let $K$ be a field of characteristic $p$ with trivial $\mathbb{G}(\mathbb{Z}_p/p^n\mathbb{Z}_p)$-action. Cohomology-ring stabilization conjecture. There exists a natural number $f=f(p,\mathbb{G})$, depending only on $p$ and $\mathbb{G}$, such that for all $n\geq f$, the cohomology rings

$$\operatorname{H}^{\bullet}(\mathbb{G}(\mathbb{Z}_p/p^n\mathbb{Z}_p);K)$$

are isomorphic. The conjecture is motivated by the fact that the Quillen categories of the groups $\mathbb{G}(\mathbb{Z}_p/p^n\mathbb{Z}_p)$ are isomorphic, while stabilization of the full cohomology-ring structure remains to be established in this generality.