Peterson's hit problem conjecture

Let $P_h$ be the polynomial algebra on $h$ variables over $\mathbb F_p$, equipped with the action of the Steenrod algebra $\mathscr A_p$. For a positive integer $n$, write its $p$-adic expansion as $n=\sum_{j\geq 0}\alpha_j(n)p^j$ with $0\leq\alpha_j(n)\leq p-1$, and set $\alpha(n)=\sum_{j\geq 0}\alpha_j(n)$. Peterson's conjecture. Every homogeneous polynomial of degree $n$ in $h$ variables is in the image of the Steenrod algebra if and only if $\alpha(n+h)>h$. Over $\mathbb F_2$ this conjecture was proved by Wood, and later for the symmetric algebras corresponding to classifying spaces of orthogonal groups by Janfada and Wood; however, it is not true for odd primes in general.