Zagier's polylogarithm conjecture for motivic Lie coalgebras

Let $F$ be a number field. For $x\in F\setminus\{0,1\}$ and $n\geq 1$, let $\operatorname{Li}_n^{\mathcal{C}}(x)\in\mathcal{C}_n(F)$ be the motivic polylogarithm, where $\mathcal{C}_n(F)$ is the degree-$n$ part of the motivic Lie coalgebra modulo products. An element is primitive when its cobracket vanishes. Zagier's polylogarithm conjecture. The space of primitive elements in $\mathcal{C}_n(F)$ is spanned by $\mathbb{Q}$-linear combinations of the elements $\operatorname{Li}_n^{\mathcal{C}}(x)$ for $x\in F\setminus\{0,1\}$. This predicts that classical motivic polylogarithms capture the primitive part of the motivic Lie coalgebra, and hence the corresponding algebraic K-theory, but the source does not specify a resolution.