Representable-functor conjecture for cohomology of integral nilpotent group schemes

Let $G$ be a nilpotent affine group scheme over the integers, and let $A$ be a representable functor of commutative graded $\mathbb{Z}$-algebras. Integral group-scheme representability conjecture. There exists such a functor $A$ such that, for almost all primes $p$,

$$\operatorname{H}^{*}(G(\mathbb{Z}_p);\mathbb{F}_p)\cong A(\mathbb{F}_p)\quad\text{and}\quad \operatorname{H}^{*}(G(\mathbb{Z}_p);\mathbb{Q}_p)\cong A(\mathbb{Q}_p).$$

This extends the preceding conjecture from saturable nilpotent pro-$p$ groups to integral points of nilpotent affine group schemes, with the comparison required for all but finitely many primes; the paper gives no general proof or disproof.