Affineness and affinization of the Eilenberg–Mac Lane stack of the unipotent completion

For an integer $n$, let $\bbH$ be the affine group scheme of formal series $P(X)\in 1+XA[[X]]$ satisfying $P(X)P(Y)=P(X+Y+XY)$. The assignment $m\mapsto(1+X)^m$ defines a morphism of presheaves of groups $\mathbb{Z}\longrightarrow\mathbb{H}$ and hence a morphism of simplicial presheaves $K(\mathbb{Z},n)\longrightarrow K(\mathbb{H},n)$. Affineness conjecture. The stack $K(\mathbb{H},n)$ is affine, and the morphism

$$K(\mathbb{Z},n)\longrightarrow K(\mathbb{H},n)$$

previously defined is an affinization. This predicts that the simplicial classifying object associated with the unipotent completion of $\mathbb{Z}$ is affine and universally captures the affine approximation of $K(\mathbb{Z},n)$.