Let $k$ be a field (of char. not $2$) and $X_k=\text{Spec} (k[x_1,\cdots,x_n]/(x_1^2+\cdots +x_n^2-1))$. Do we know the Chow groups $A_i (X_k)$? I could not find any references, even for $X_{\mathbb R}$.

What (I think) I know: the K-groups were computed by Swan, so we know the total Chow group up to torsions. In codimension $1$ (i.e., class groups) I am fairly certain the answers are known.