@HailongDao, I too am waiting for him, like Gogo for Godot.

@Denis-CharlesCisinski, I too had in mind to think of [𝔾𝑚,𝑆] as a kind of loop space and pointed-maps version of $[𝔾𝑚,𝑆]_{\mathbb{A}^1}$ as a kind of fundamental group. Does that makes some sense? Using $\mathbb{G}_m^n$, or for that matter $\mathbb{P}^n$, as probe spaces to explore the topology of $S$ just as one does with spheres $S^n$ in classical homotopy theory seems like a fruitful idea. Would love your comments on it.