Classical blue-shift phenomenon

Let $G$ be a finite group and let $E$ be a $v_n$-periodic non-equivariant ring spectrum. The classical Tate construction is

$$t_{G}({\rm \inf}^G_{e}(E))^G.$$

Classical blue-shift phenomenon. The spectrum $t_{G}({\rm \inf}^G_{e}(E))^G$ is $v_{n-s_{G;E}}$-periodic for some positive integer $s_{G;E}$. When $s_{G;E}>n$, the $v_{n-s_{G;E}}$-periodic ring spectrum is understood to be the contractible spectrum $*$. The integer $s_{G;E}$ is called the blue-shift number. This conjecture generalizes known blue-shift and Tate-vanishing results, including the case of the group $\mathbb{Z}/2$ acting on connective complex K-theory; its general validity is not established.