Goncharov's weight-four element conjecture

Let $E$ be a field and let $x\neq y$ and $z$ be elements of $E\setminus\{0,1\}$. Define

$$A(x,y)=[x]-[y]-[x/y]+[(1-x)/(1-y)]-[(1-1/x)/(1-1/y)].$$

Let $[x,z]_{3,1}$ denote the weight-four hyperlogarithmic symbol used in the paper, and set

$$B(x,y;z)=[x,z]_{3,1}-[y,z]_{3,1}-[x/y,z]_{3,1}+[(1-x)/(1-y),z]_{3,1}-[(1-1/x)/(1-1/y),z]_{3,1}.$$

Goncharov's conjecture. The element $B(x,y;z)$ belongs to ${\cal P}_4(E)$; equivalently, it is a linear combination of $4$-logarithms. The construction gives an element killed by $\delta_{2,2}$, and the conjecture is a concrete instance of the proposed weight-four reduction from hyperlogarithms to polylogarithms.