Generalized-cohomology conjecture for invertible topological phases

For invertible phases, let $\Theta_d$ be the space appearing in the internal- and spatial-symmetry classification conjectures. Let $\Omega\Theta_{d+1}$ denote the based loop space consisting of maps $\gamma:[0,1]\to\Theta_{d+1}$ satisfying $\gamma(0)=\gamma(1)=\vartheta_*$, where $\vartheta_*\in\Theta_{d+1}$ is a basepoint representing the trivial vacuum state. Write $\simeq$ for homotopy equivalence.

Generalized-cohomology conjecture. The spaces $\Theta_d$ can be chosen to satisfy

$$\Theta_d\simeq\Omega\Theta_{d+1}.$$

This assumption supplies the generalized-cohomology structure used for invertible crystalline phases and is attributed in the paper to earlier work. No resolution is given in the supplied text.