Koszulity conjecture for isotropic Milnor K-rings

Let $k$ be a flexible field and let $F/k$ be a finitely generated field extension. The isotropic Milnor K-ring of $F/k$, denoted $\operatorname{\mathrm{k}}^M_*(F/k)$, is the quotient of the tensor algebra on $\operatorname{\mathrm{k}}^M_1(F/k)$ over $\operatorname{{\mathbb F}}_2$ by the ideal generated by the elements $a\otimes(1-a)$. Koszulity conjecture. For every finitely generated field extension $F/k$, the isotropic Milnor $K$-ring $\operatorname{\mathrm{k}}^M_*(F/k)$ is a Koszul $\operatorname{{\mathbb F}}_2$-algebra. This would, by the preceding discussion, identify the cellular isotropic $\operatorname{\mathrm{MBP}}$-module category over $F/k$ with the derived category of the heart of its motivic $t$-structure. It is not known in general whether these isotropic Milnor K-rings are Koszul.