Goncharov's Aomoto–Grassmannian comparison conjecture

Let $\widetilde A_n$ be the free abelian group generated by admissible pairs of simplices, let $\overline C_{2n}$ be the Grassmannian configuration group, and suppose the maps $a_i$ and the cobracket $\delta_n$ have been defined in lower weights. Aomoto–Grassmannian comparison conjecture. There exists a homomorphism

$$a_n:\widetilde A_n\to\overline C_{2n}$$

such that the diagram involving the map $\nu:\widetilde A_n\to\bigoplus_{1\leq i\leq n}A_i\otimes A_{n-i}$ and the maps $a_i\wedge a_{n-i}$ and $\delta_n$ commutes, and such that

$$\mathcal L_n(L;M)=c_n\mathcal L_n^G(a_n(L,M))$$

for every $(L;M)\in A_n(\mathbb C)$, where $c_n$ is a normalization constant. This would relate Aomoto polylogarithms to Grassmannian polylogarithms and make the corresponding motivic cobracket compatible.