Internal-symmetry homotopy classification conjecture for topological phases

Let $G$ be an internal symmetry group, let $BG$ be its classifying space, and let $\Theta_d$ denote the space classifying $d$-dimensional topological phases. The classifying space is, up to homotopy equivalence, $BG=EG/G$, where $EG$ is any contractible space with a free action of $G$.

Internal-symmetry classification conjecture. The classification of SPT or SET topological phases in $d$ dimensions with internal symmetry $G$ is given by homotopy classes of maps

$$f:BG\to\Theta_d.$$

The paper presents this as a starting assumption and notes that it has appeared previously in several forms. Its resolution is not stated in the supplied text.