Compatibility with global classifying spaces of finite groups

Let $G$ be a finite group. Write $\mathbf B G\in\mathrm{OrbSpc}$ for its classifying orbispace and $\mathbf B G\in\mathrm{GloSpc}$ for the corresponding global classifying space. Classifying-space compatibility conjecture. The functor from orbispaces to global spaces sends $\mathbf B G\in\mathrm{OrbSpc}$ to $\mathbf B G\in\mathrm{GloSpc}$. This is a basic compatibility requirement for the conjectural identification of orbispaces with their associated global spaces.