The logarithmic Milnor K-theory birationality conjecture

Let $\mathbf{k}$ be a field of characteristic $0$. For a field extension $F/\mathbf{k}$, write

$$K_2(F)/\mathrm{Const}=K^M_2(F)\mathbin{/}\left\langle\{F^*,\mathbf{k}^*\}\right\rangle.$$

A dominant rational map induces pullback on Milnor $K$-groups. The logarithmic Milnor K-theory conjecture. If $\varphi:\mathbb{A}^2\dashrightarrow\mathbb{A}^2$ is dominant and

$$\varphi^*\{x,y\}=\{x,y\}\in K_2/\mathrm{Const},$$

then $\varphi$ is birational. This is presented as another version of Galkin's conjecture, equivalent to the formulation using the logarithmic volume form; it remains open in the source.