Goncharov's classical polylogarithm depth conjecture

Let $F$ be an infinite field. The truncated cobracket

$$\overline{\delta}\colon\mathcal{L}^{\mathrm{f}}(F)\longrightarrow\bigwedge^2\mathcal{L}^{\mathrm{f}}(F)$$

is obtained by omitting the $\mathcal{L}_1(F)\wedge\mathcal{L}_{n-1}(F)$ component of the cobracket. Goncharov's classical polylogarithm conjecture. For $n\geq1$, an element $x\in\mathcal{L}^{\mathrm{f}}_n(F)$ is a linear combination of classical polylogarithms $\operatorname{Li}^{\mathcal{L}}_n(a)$ for $a\in F^{\times}$ if and only if $\overline{\delta}(x)=0$. The vanishing direction follows from the preceding lemma, while the converse is conjectural in general.