On a relation of a conjecture of Goncharov to the co-Lie algebra of Bloch-Kriz mixed Tate motives

Goncharov defined for each field $F$ and an integer $n$ greater than 1 a certain group $B_n(F)$. We consider the possibility of defining a linear map from $B_n(F)$ to the co-Lie algebra of the category of mixed Tate motives defined by Bloch and Kriz, in terms of motivic polylogarithms. We give results which support this possibility assuming part of the conjecture by Beilinson and Soulé on vanishing of $K$-groups of fields.