The dimension conjecture for the kernel of Kameko's squaring operation

Let $P_k=\mathbb F_2[x_1,\ldots,x_k]$ be the polynomial algebra, let $Q P_k$ denote its quotient by the image of the positive-degree Steenrod algebra, and let

$$(\widetilde{Sq}^{0}_*)_{(k,n)}:(Q P_k)_n\longrightarrow (Q P_k)_{(n-k)/2}$$

be Kameko's squaring operation for $n-k$ even. Let $k\geq 3$ and write

$$n=\sum_{i=1}^{k-2}(2^{d_i}-1),$$

where the $d_i$ are positive integers. The dimension conjecture for the kernel of Kameko's squaring operation. If $d_{i-2}-d_{i-1}>i$ for $3\leq i\leq k-1$ and $d_{k-2}>k$, then

$$\dim\operatorname{Ker}(\widetilde{Sq}^{0}_*)_{(k,n)}=\prod_{i=3}^{k}(2^i-1).$$

This predicts the dimension of the kernel in the degree range with $\mu(n)=k-2$ and is intended to describe the unresolved hit problem for the polynomial algebra in these generic degrees.