The Evenness Conjecture for Real equivariant homotopy groups

Let $G$ be a compact quasi-abelian Lie group equipped with an augmentation

$$\eta:G\longrightarrow C_2.$$

Write $\rho$ for the relevant regular representation and $M\mathbb{R}_\eta$ for the associated Real equivariant spectrum. Evenness Conjecture. For every such augmented group,

$$\pi^G_{*\rho-1}(M\mathbb{R}_\eta)=0.$$

When $\eta$ is trivial, this reduces to the known evenness of $MU_G^*$ for abelian compact Lie groups. The conjecture would imply that the split surjection from the Real global group law to the ordinary global group law is an isomorphism; it is proposed as an ingredient for proving the Hu–Kriz map result and remains open in the source.