Extension and compatibility conjecture for equivariant homology of spectra

Let $G$ be a finite group. Write $\operatorname{\sf StHom}^{naive}(G)$ and $\operatorname{\sf StHom}(G)$ for the categories of naive and genuine $G$-spectra, respectively, and let $i_!$ be the tautological functor between them. Let $h^G_{naive}$ and $h^G$ be the naive and equivariant homology functors, let $q^{opp}_!$ be the indicated functor between the derived categories of coefficient systems and derived Mackey functors, and let $\Phi^H$ and $\Phi^{[G/H]}$ denote the geometric fixed-point functors on spectra and derived Mackey functors, respectively. Extension and compatibility conjecture. The functors $h^G$ and $h^G_{naive}$ extend to the categories of spectra so that the diagram

$$\begin{CD} \operatorname{\sf StHom}^{naive}(G) @>{i_!}>> \operatorname{\sf StHom}(G)\\ @V{h^G_{naive}}VV @VV{h^G}V\\ {\cal D}(O_G^{opp},{\mathbb Z}{\text{-mod}}) @>{q^{opp}_!}>> {\cal D}{\cal M}(G) \end{CD}$$

is a commutative diagram of tensor triangulated functors. Moreover, for every subgroup $H\subset G$, there is an isomorphism of functors

$$\Phi^{[G/H]}\circ h^G(X)\cong h\circ\Phi^H(X)$$

from $\operatorname{\sf StHom}(G)$ to ${\cal D}({\mathbb Z}{\text{-mod}})$. The conjecture extends the finite-spectrum compatibility to all spectra and predicts compatibility with geometric fixed points; the source provides no evidence of resolution, so its status remains open.