Compatibility of the global-space functor with the presheaf equivalence

Let $\mathrm{OrbSpc}$ be the category of orbispaces, let $\overline{\mathrm{OrbSpc}}$ be its enlargement obtained by gluing cells under arbitrary maps, and let $\mathrm{GloSpc}$ be the category of global spaces. The paper considers an equivalence

$$\overline{\mathrm{OrbSpc}}\simeq\mathrm{GloSpc}$$

constructed from results of Gepner--Henriques and Schwede. Global-space compatibility conjecture. The restriction of this equivalence to the full subcategory $\mathrm{OrbSpc}\subseteq\overline{\mathrm{OrbSpc}}$ coincides with the functor sending an orbispace $X$ to $\operatorname{Emb}_X(E,-)$ for a faithful vector bundle $E$ over $X$. This would identify the geometric global-space construction with the presheaf-theoretic equivalence.