The Steenrod-algebra kernel-image conjecture

Let $\mathcal{A}$ be the mod-$2$ Steenrod algebra, let $\beta$ denote the relevant operation, and consider the quotient left $\mathcal{A}$-module $\mathcal{A}/\mathcal{A}\beta$. For each $k\geq 0$, examine the kernel and image of the indicated Steenrod operations in degrees at most $8k+1$. Kernel-image conjecture. In $\mathcal{A}/\mathcal{A}\beta$, one has

$$\ker \operatorname{Sq}^{8k+1}=\operatorname{im}\operatorname{Sq}^{4k+1}$$

in degrees up to $8k+1$, for all $k\geq 0$. This is the basic calculation needed for the spectral sequence computing strict units of the polynomial ring $H[u]$. The supplied text says that the conjecture was verified by computer for small values of $k$, while the general case remains open.