Kameko's dimension conjecture for the Peterson hit problem

Let $P_k$ be the polynomial algebra on $k$ generators over $\mathbb{F}_2$, let $QP_k$ denote its indecomposable quotient under the Steenrod algebra, and write $(QP_k)_n$ for its degree-$n$ component. Kameko's conjecture. For every non-negative integer $n$,

$$\dim (QP_k)_n \leqslant \prod_{1\leqslant i \leqslant k} (2^i-1).$$

This conjecture asserts a uniform bound on the dimensions of the graded components of the indecomposable quotient, depending only on $k$; the supplied source does not state a resolution.