The equivariant generating hypothesis for finite G-spectra

Let $G$ be a compact Lie group, let $\mathcal{M}[G]$ be the category of Mackey functors over $G$, and let $\mathrm{Ho}G\mathcal{S}$ be the homotopy category of $G$-spectra. The equivariant homotopy functor

$$\underline{\pi}_*^G(-):\mathrm{Ho}G\mathcal{S}\to\mathcal{M}[G]$$

records the equivariant homotopy groups.

Equivariant generating hypothesis. The restriction of $\underline{\pi}_*^G(-)$ to the subcategory of finite $G$-spectra is faithful. Equivalently, if a map $f:X\to Y$ between finite $G$-spectra induces the zero map $f_*:\underline{\pi}^G_*(X)\to\underline{\pi}_*^G(Y)$, then $f$ is nullhomotopic.

This is the proposed equivariant generalization of Freyd's generating hypothesis, incorporating homotopy groups of fixed-point spectra through Mackey functors. Its status is not resolved in the supplied text; the paper proves implications for finite groups and gives a counterexample in rational $S^1$-equivariant spectra.