Bloch–Quillen formula for the special fiber

Let $X$ be the reduced special fiber of the scheme considered in the paper, of dimension $d$, and let $\mathcal{K}^M_{d,X}$ be the Nisnevich Milnor $K$-theory sheaf in degree $d$. The cycle class map is

$$\operatorname{cyc}_X^M\colon \operatorname{CH}_0^{LW}(X)\twoheadrightarrow H^d(X_{\rm Nis},\mathcal{K}^M_{d,X}).$$

Bloch–Quillen formula. The cycle class map $\operatorname{cyc}_X^M$ is an isomorphism. Together with the known surjectivity, this would identify the Levine–Weibel Chow group of zero-cycles with the relevant Nisnevich cohomology group and would provide the descent needed in the restriction diagram from the total space to the special fiber.