Higher Gangl formula for motivic multiple polylogarithms

Let $F$ be the field under consideration, let $k$ be a positive integer, and let $\mathcal{G}_k$ be the quotient of $\operatorname{gr}_k^\mathcal{D}\mathcal{L}_{2k}(F)$ by the subspace spanned by the higher Zagier formulae. Let $\mathcal{B}_2(F)$ be the weight-two Bloch group. Higher Gangl formula. The map

$$\overline{\Delta}^{[k-1]}\colon\mathcal{G}_k\longrightarrow\operatorname{CoLie}_k(\mathcal{B}_2(F))$$

is an isomorphism. The source records this claim as resolved, with the cited discussion indicating that the first cases are established and that the relevant formula is proved in the paper's context.