Picard-space identification

Establish or refute the equivalence \(\mathrm{Pic}(P_2\mathcal{S}_*) \simeq \Omega^\infty \mathbb{RP}^\infty_{-1}\).

Background

The paper proves that the Picard group of the metastable category is trivial and computes $$\pi_1\mathrm{Pic}(P_2\mathcal{S}_*)\cong\mathbb{Z}/2$$. It also identifies the loop spaces of exotic 0-spheres using the spectrum $\mathbb{RP}^\infty_{-1}$.

These results suggest that the entire Picard space may admit a natural description in terms of the infinite loop space of $\mathbb{RP}^\infty_{-1}$, but the proposed equivalence is not established.