  Let $X$ be a smooth proper variety over a field $k$ and suppose that the
degree map $\mathrm{CH}_0(X \otimes_k K) \to \mathbb{Z}$ is isomorphic for any
field extension $K/k$. We show that $G(\mathrm{Spec} k) \to G(X)$ is an
isomorphism for any $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers
$G$. This generalize a result of Binda-R\"ulling-Saito that proves the same
conclusion for reciprocity sheaves. We also give a direct proof of the fact
that the unramified logarithmic Hodge-Witt cohomology is a
$\mathbb{P}^1$-invariant Nisnevich sheaf with transfers.
