The multiplicative tom Dieck splitting conjecture for structure Xi

Let $G$ be a compact Lie group, let $K$ range over the closed subgroups of $G$, and write $W_K=N_G(K)/K$ for the Weyl group and $\Ad_{W_K}$ for its adjoint representation. Let $R$ be an augmented $E_{\infty}$ algebra, let $R^{\Phi K}$ denote its geometric $K$-fixed points, and let $j_{\mathcal{E}}^{\Xi}$ denote the comparison from the putative multiplicative structure $\Xi$ to the operadic structure. The notation $R^{\Phi K}\mathbin{\widehat\otimes}S^{\Ad_{W_K}}$ denotes the dimension-shifted based tensor, and $(-)\otimes_{W_K}EW_K$ denotes the Weyl-group orbit construction.

Multiplicative tom Dieck splitting conjecture. There exists a multiplicative structure $\Xi$ such that $\Xi$-algebras are in particular $G$-$E_{\infty}$ algebras and, in the augmented or nonunital setting, admit dimension-shifting multiplicative transfers for all closed subgroups. For every augmented $E_{\infty}$ algebra $R$, these transfers induce an equivalence

$$\coprod_{(K)<G} \$R^{\Phi K} \mathbin{\widehat\otimes} S^{\Ad_{W_K}}\$\otimes_{W_K}EW_K \xrightarrow{\\,\sim\\,} ((j_{\mathcal{E}}^{\Xi})_{*}R)^{\Phi G}.$$

The conjecture proposes a multiplicative enhancement of the additive tom Dieck splitting, extending the observed dimension-shifting transfer structure beyond ordinary operadic algebras. Its status is unknown: the existence of $\Xi$ and the asserted splitting remain to be established.