The modular invariant-theory conjecture for arbitrary representations

Let $G\subset GL(V)$ be a finite group, let $F$ be a field, and let $V$ be a finite-dimensional $F$-vector space. Let $S(V)$ be the symmetric algebra of $V$, and write $S(V)^G$ for its invariant subring. For a one-dimensional subspace $U\subset V^*$, let $G_U=\{g\in G\mid g(u)=u\text{ for all }u\in U\}$. Modular invariant-theory conjecture. $S(V)^G$ is a polynomial ring if and only if:

1. $G$ is generated by pseudo-reflections; and
2. $S(V)^{G_U}$ is a polynomial ring for each subspace $U\subset V^*$ with $\operatorname{dim}_F U=1$.

This conjecture proposes that the necessary conditions known for arbitrary representations are also sufficient, extending the Kemper–Malle theorem beyond the irreducible case. The source states that the corresponding sufficiency question remains open in the modular case.