Voevodsky's $S^1$-connectedness conjecture

Let $k$ be the base field and let ${\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k)$ be the Morel–Voevodsky homotopy category of $S^1$-spectra. For an integer $d$, let $\Sigma^d_{{\mathbb P}^1}{\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k)$ denote the localizing subcategory generated by the presheaves $\Sigma^d_{{\mathbb P}^1}F$. Voevodsky's $S^1$-connectedness conjecture. If $E$ belongs to $\Sigma^d_{{\mathbb P}^1}{\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k)$, then

$$\Omega_{{\mathbb P}^1}\Sigma_{{\mathbb P}^1}E$$

also belongs to $\Sigma^d_{{\mathbb P}^1}{\operatorname{{\mathcal S}{\mathcal H}}}_{S^1}(k)$. This is the motivic analogue of the fact that $\Omega\Sigma X$ preserves connectivity. The paper states that the result follows from its identification of the $S^1$-slice tower with the homotopy coniveau tower, so the conjecture is resolved in the stated setting.