Compatibility with the global classifying space of stable vector bundles

Let $\mathbf{bO}$ denote the global space that is the direct limit of the global classifying spaces $\mathbf B O(n)$, and regard $\mathbf{bO}$ also as an orbispace through the functor under consideration. Stable-vector-bundle classifying-space conjecture. The functor from orbispaces to global spaces sends $\mathbf{bO}\in\mathrm{OrbSpc}$ to $\mathbf{bO}\in\mathrm{GloSpc}$. This would extend the compatibility with the global classifying spaces $\mathbf B G$ to the global space classifying rank-zero coarsely stable vector bundles.