Goncharov's Hodge multiple-zeta Hopf algebra conjecture

Let ${\cal Z}^{\cal H}_{\bullet}$ be the graded Hopf algebra of framed Hodge-Tate structures generated by the Hodge multiple-zeta structures, and let $U{\cal F}(3,5,...)_{\bullet}^{\vee}$ be the graded dual of the universal enveloping algebra of the free graded Lie algebra generated in degrees $-(2n+1)$ for $n\geq 1$. Goncharov's conjecture. There exists an isomorphism of graded Hopf algebras over $\mathbb Q$:

$${\cal Z}^{\cal H}_{\bullet}\cong U{\cal F}(3,5,...)_{\bullet}^{\vee}.$$

This is the Hodge-theoretic version of the conjectural description of the algebra of multiple zeta values; the paper notes that it implies the corresponding dimension and relation statements.