Lifting the computed loop-space equivalence to the metastable category

Establish or refute the equivalence \(\Omega S^1 \simeq \mathbb{Z} \times \Omega^{\infty+2}\mathbb{Sp}^2\) in the metastable category \(P_2\mathcal{S}_*\).

Background

A companion result identifies the corresponding loop space after applying the 2-excisive approximation to spaces: $$\Omega P_2I(S^1)\simeq \mathbb{Z}\times\Omega^{\infty+2}\mathbb{Sp}^2$$. The paper also proves a 2-local equivalence for $\Omega^2S^1$ in the metastable setting.

What remains unresolved is whether the one-fold loop-space equivalence itself can be realized inside the metastable category. The obstruction is the absence of an evident map from $\Omega^\infty\Sigma\mathbb{Z}$ to $S^1$ in $P_2\mathcal{S}_*$.