The irreducible-object conjecture for $\mathcal I_G$

Let $F/k$ be the field extension in the source, let $G$ be its field automorphism group, and let $\mathcal I_G$ be the category of homotopy-invariant smooth $G$-representations. Let $\Omega^{\bullet}_{F/k}$ denote the graded algebra of differential forms of $F$ over $k$. Irreducible-object conjecture. If $n=\infty$, then every irreducible object of $\mathcal I_G$ is contained in the algebra $\Omega^{\bullet}_{F/k}$. This would constrain the irreducible homotopy-invariant representations by differential-form representations; the source gives no resolution.