Stable SPT classification group-structure conjecture

Let $S$ be the set of stable SPT classes for a fixed group and fixed lattice. For stable classes $\expval{\omega_1,U_1}_{\sim}$ and $\expval{\omega_2,U_2}_{\sim}$, define

$$*(\expval{\omega_1,U_1}_{\sim},\expval{\omega_2,U_2}_{\sim})=\expval{\omega_1\otimes_{\mathrm{stack}}\omega_2,U_1\otimes U_2}_{\sim}.$$

Stable SPT classification group-structure conjecture. The operation $*$ is well defined, independently of the representatives, and makes $S$ an abelian group.

The conjecture formalizes the expected group structure supplied by stacking stable SPT classes; the existence of inverses is related to $G$-inverse states. The source does not provide a resolution, so the conjecture remains open.