Carlisle–Kuhn polynomiality conjecture for the Grothendieck ring of unstable modules

Let $K^n(\operatorname{\mathcal{U}})$ be the free abelian group generated by isomorphism classes of direct summands of $H^*V_n$, where $V_n=(\operatorname{\mathbb{F}}_p)^n$, and let

$$K(\operatorname{\mathcal{U}})=\bigcup_{n\geq 0}K^n(\operatorname{\mathcal{U}}).$$

The ring structure is induced by tensor products of unstable modules. Carlisle–Kuhn's conjecture. The ring $K(\operatorname{\mathcal{U}})$ is polynomial over $\mathbb{Z}$. The conjecture concerns the algebraic structure of Grothendieck groups arising from summands of cohomology modules over the Steenrod algebra; the supplied source does not state a resolution, so its status is recorded as open.