Kameko's dimension conjecture for the hit problem

Let $E^k$ be an elementary abelian $2$-group of rank $k$, let $BE^k$ be its classifying space, and set

$$P_k:=H^*(BE^k)=\mathbb F_2[x_1,x_2,\ldots,x_k].$$

Write $\mathbb F_2\underset{\mathcal A}\otimes P_k$ for the quotient of $P_k$ by the positive-degree Steenrod operations, and let $(\mathbb F_2\underset{\mathcal A}\otimes P_k)_n$ denote its degree-$n$ component. Let $s_1,\ldots,s_{k-1}$ be positive integers and define

$$n=2^{s_1+s_2+\ldots+s_{k-1}}+2^{s_1+s_2+\ldots+s_{k-2}}+\ldots+2^{s_1}-k+1.$$

If $s_j\geqslant2$ for all $j$, then Kameko's conjecture.

$$\dim\left(\mathbb F_2\underset{\mathcal A}\otimes P_k\right)_n=\prod_{1\leqslant i\leqslant k}(2^i-1).$$

The paper proves the related supremum formula for $k=4$ using an explicit determination of the corresponding quotient. The displayed conjecture is stated for general $k$; no resolution is supplied in the given text.