Goncharov's Grassmannian Lie coalgebra and mixed Tate motives conjecture

Let $F$ be a field and let

$$G_\bullet(F):=\bigoplus_{n\geq1}G_n(F).$$

The maps

$$\delta_n:G_n(F)\to\bigoplus_{i\leq n/2}G_i(F)\wedge G_{n-i}(F)$$

are required to make the induced sequence $G_\bullet(F)\to\Lambda^2G_\bullet(F)\to\Lambda^3G_\bullet(F)\to\cdots$ a complex. Goncharov's motivic Lie coalgebra conjecture. The graded vector space $G_\bullet(F)$ has a natural structure of a graded Lie coalgebra over $\mathbb Q$; the category of graded finite-dimensional $G_\bullet(F)$-modules is equivalent to the category of mixed Tate motives over $\operatorname{Spec}F$, and

$$H^i_{(n)}\bigl(G_\bullet(F)\bigr)=\operatorname{gr}^{\gamma}_nK_{2n-i}(F)\otimes\mathbb Q.$$

The conjecture proposes that the Grassmannian construction gives the motivic Lie coalgebra governing mixed Tate motives and recovers the corresponding Adams-graded $K$-theory groups.