The motivic connectivity conjecture for $\mathbb{P}^1$-spectra

Let $k$ be the base field, let ${\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k)$ be the homotopy category of $S^1$-spectra, and let $\Sigma^\infty_{{\mathbb P}^1}$ and $\Omega^\infty_{{\mathbb P}^1}$ be the adjoint $\mathbb{P}^1$-suspension-spectrum and infinite-loop functors. The motivic connectivity conjecture. If

$$E\in\Sigma_{{\mathbb P}^1}^n{\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k),$$

then

$$\Omega_{{\mathbb P}^1}^\infty\Sigma^\infty_{{\mathbb P}^1}E\in\Sigma_{{\mathbb P}^1}^n{\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k).$$

This is the corresponding connectivity assertion for $\mathbb{P}^1$-spectra. The supplied text states it as a conjecture but gives no resolution, so its status is left open.