Existence of the algebra of higher cohomology operations

Let $R=\mathbb{Z}/p^2$, and let $\mathcal{M}^N$ be the category of finitely generated free graded $R$-modules concentrated in degree less than $N$. For a bigraded differential algebra $Q$ over $R$, write $Q(n)$ for its truncation at order $n$, and let ${\mathbf{C}}^n\{\mathcal{Z}_N\}$ denote the stable track category of higher cohomology operations. Existence conjecture. There exists a bigraded differential algebra $Q$ over $R$ such that, for every $n\geq 0$, the track category

$$\left({\mathbf{K}}^{Q(n)}(\mathcal{M}^N)\right)^{\mathrm{op}}$$

is weakly equivalent to ${\mathbf{C}}^n\{\mathcal{Z}_N\}$. This $Q$ is called the algebra of higher cohomology operations. The claim would provide an algebraic model for stable higher cohomology operations and relate their higher track-category structure to differential-algebraic data.