Global refinement conjecture for tempered and equivariant elliptic cohomology

Let ${\mathscr{F}}$ be a geometric oriented abelian sheaf over a derived stack $\mathsf{S}$. Write $\operatorname{Sp}_{\mathrm{fin}-\mathrm{gl}}$ for the symmetric monoidal $\infty$-category of global spectra based on finite groups, and let

$$\mathrm{res}_\mathrm{fin}\colon \operatorname{Sp}_\mathrm{gl}\to\operatorname{Sp}_{\mathrm{fin}-\mathrm{gl}}$$

be the restriction functor. Let $\Gamma_\mathrm{gl}\mathrm{Ell}_{{\mathscr{F}}/\mathsf{S}}$ and $\Gamma_{\mathrm{fin}-\mathrm{gl}}\mathrm{Temp}_{{\mathscr{F}}[\mathbf{P}^\infty]/\mathsf{S}}$ denote the global refinements of equivariant elliptic and tempered cohomology, respectively. Global comparison conjecture. The two $\mathbf{E}_\infty$-objects in $\operatorname{Sp}_{\mathrm{fin}-\mathrm{gl}}$,

$$\mathrm{res}_\mathrm{fin}\Gamma_\mathrm{gl}\mathrm{Ell}_{{\mathscr{F}}/\mathsf{S}}$$

and

$$\Gamma_{\mathrm{fin}-\mathrm{gl}}\mathrm{Temp}_{{\mathscr{F}}[\mathbf{P}^\infty]/\mathsf{S}},$$

are naturally equivalent. This would enhance the comparison between tempered and equivariant elliptic cohomology from agreement on finite-group fixed points to an equivalence of their global finite-group refinements; the claim is presented as a conjectural refinement, and no resolution is given in the source.