Morel's stable $\mathbb{A}^1$-connectivity conjecture

Let $S$ be a regular noetherian scheme, and let $n\geqslant 0$ be an integer. For any $n$-connective pointed space $E\in\mathcal{P}_{\operatorname{Nis}}(\operatorname{Sm}_S)_*$, the motivic spectrum $L_{\operatorname{mot}}E\in\operatorname{SH}(S)$ is $n$-connective.

Morel's stable $\mathbb{A}^1$-connectivity conjecture. Under these hypotheses, $L_{\operatorname{mot}}E$ is $n$-connective.

This conjecture concerns preservation of connectivity by motivic localisation in the stable setting. It is refuted in higher-dimensional cases: Ayoub exhibited counterexamples when $S$ is a scheme of higher dimension and explained why regularity should not control stable $\mathbb{A}^1$-connectivity.