Goncharov's five-term relation conjecture for the symbol of $I_{3,1}$

Let $V(x,y)$ denote any version of the five-term relation, and let $I_{3,1}(V(x,y),z)$ be the corresponding weight-four multiple polylogarithmic expression. Let $f_j(x,y,z)$ be rational functions in the three variables $x$, $y$, and $z$. Write ${\mathcal S}$ for the symbol map, and interpret the equality modulo products.

Goncharov's conjecture. There are rational functions $f_j(x,y,z)$ such that

$${\mathcal S}\big(I_{3,1}(V(x,y),z)\big)={\mathcal S}\bigg(\sum_j \operatorname{Li}_4\big(f_j(x,y,z)\big)\bigg).$$

The conjecture reformulates Goncharov's expected cohomological vanishing for the weight-four thickened motivic complex and predicts that the relevant $I_{3,1}$ symbol is expressible using classical 4-logarithms, modulo products. The source gives no evidence of resolution.