Embedding of category of twisted Chow-Witt correspondences into geometric stable ${\text{A}}^1$-derived category over a field

We introduce in this note the notion of the category of twisted Chow-Witt correspondences $CHW(k)$ over a field $k$ of characteristic different from $2$. Moreover, we show that over an infinite perfect field this category $CHW(k)$ admits a fully faithful embedding into the geometric stable ${\text{A}}^1$-derived category $D_{{\text{A}}^1,gm}(k)$ after taking $\text{Q}$-localization. We also prove a conjecture of F. Morel about the rational splitting of stable ${\text{A}}^1$-cohomology over an essentially smooth scheme $S$ over a field $k$ of $char(k)\mathrm{\ne }2$.