The Local Realization Conjecture for finite-degree Steenrod modules

Let $M$ be an $\mathcal A$-module concentrated in finitely many degrees, where $\mathcal A$ is the mod 2 Steenrod algebra. Local Realization Conjecture. For all sufficiently large integers $k$, there is no space or spectrum $X$ such that

$$H^*(X;\mathbb Z/2)\simeq M\otimes\Phi(k,k+1).$$

This is described as a stronger qualitative form of the finite-generation conjecture and as a restatement of the Local Realization Conjecture from the author's earlier work. The source provides no resolution.