Goncharov's cyclotomic period conjecture

Let $N$ be a positive integer. Let ${\cal Z}^{\cal H}_{\bullet}(\mu_N)$ be the commutative graded Hopf algebra generated by framed Hodge-Tate structures associated with multiple polylogarithms at $N$-th roots of unity. Let $\widetilde{\cal Z}_w(\mu_N)$ be the $\mathbb Q$-vector space spanned by $(2\pi i)^{-w}$ times weight-$w$ multiple-polylogarithm values at $N$-th roots of unity, and let $\widetilde{\cal Z}(\mu_N)$ be the resulting weight-filtered algebra. Goncharov's cyclotomic period conjecture. There is an isomorphism of commutative algebras

$$\operatorname{Gr}^W_{\bullet}\widetilde{\cal Z}(\mu_N)\cong {\cal Z}^{\cal H}_{\bullet}(\mu_N).$$

A canonical surjection in the opposite direction is known in the paper, so the conjecture asserts injectivity as well and would identify the algebraic Hodge periods with the associated graded of the numerical period algebra.