Goncharov's weight-four Grassmannian-to-polylogarithmic complex conjecture

Let $G_4(F)$ be the weight-four Grassmannian group, let $\mathcal B_4(F)$ be the corresponding polylogarithmic group, and let

$$\overline\delta:G_4(F)\to B_3(F)\otimes F^*$$

be the canonical cobracket described in the construction. Weight-four comparison conjecture. There exists a canonical homomorphism

$$\mathbb L_4:G_4(F)\to\mathcal B_4(F)$$

that makes the diagram with $\overline\delta$ and the cobracket $\delta:\mathcal B_4(F)\to B_3(F)\otimes F^*$ commute. Such a map would compare the Grassmannian and polylogarithmic realizations in weight four.