Higher syzygy degree bounds for finite-group invariants

Let $G$ be a finite group in the nonmodular case acting on a finite-dimensional representation $V$. Let $\beta_G^i(V)$ denote the degree bound for the $i$-th syzygies of the invariant ring, and let $\tau_G(V)$ be the smallest integer $d$ such that every homogeneous polynomial of degree $d$ lies in the ideal generated by all homogeneous invariants of positive degree. Higher syzygy bound conjecture. If $G$ is a finite group in the nonmodular case, then

$$\beta_G^i(V)\leq (i+1)\tau_G(V)\leq (i+1)|G|.$$

The preceding theorem proves the corresponding bound for the syzygy ideal, but the authors state that its proof does not seem to extend to higher syzygies. The conjecture proposes analogous bounds for all higher syzygies, using Fogarty's inequality $\tau_G(V)\leq |G|$.