The improved Milnor K-theory conjecture for local rings

Let $A$ be a local ring, let $i\geq 0$ be an integer, let $\widehat{\text{K}}{}^{\text{M}}_i(A)$ denote the improved Milnor $K$-group, and let $\text{H}^i_{\text{mot}}(A,\operatorname{\mathbb{Z}}(i))$ denote motivic cohomology. The improved Milnor K-theory conjecture. The natural map

$$\widehat{\text{K}}{}^{\text{M}}_i(A) \longrightarrow \text{H}^i_{\text{mot}}(A,\operatorname{\mathbb{Z}}(i))$$

induced by Lemma~ is an isomorphism of abelian groups.

The conjecture was proved by Elmanto--Morrow for equicharacteristic local rings; the source does not establish the assertion in the full generality stated here.