Cyclotomic Lie algebra conjecture for multiple polylogarithms

Let ${\cal Z}(N)$ be the algebra of multiple polylogarithm values at $N$-th roots of unity, filtered by weight, and let $UC_{\bullet}(N)^{\vee}$ denote the graded dual of the universal enveloping algebra of a graded Lie algebra $C_{\bullet}(N)$. For $n\geq 1$, write $H^1_{(n)}$ for the weight-$n$ component of the Lie algebra cohomology, and set $S_N=\operatorname{Spec}\mathbb{Z}[\zeta_N][\frac{1}{N}]$. Cyclotomic Lie algebra conjecture. There exists a graded Lie algebra $C_{\bullet}(N)$ over $\mathbb{Q}$ such that

$${\cal Z}(N)\simeq UC_{\bullet}(N)^{\vee}$$

as algebras filtered by weight on the left and degree on the right, and such that

$$H^1_{(n)}(C_{\bullet}(N))=K_{2n-1}(\mathbb{Z}[\zeta_N][\frac{1}{N}])\otimes\mathbb{Q}.$$

Moreover, $C_{\bullet}(1)$ is a free graded Lie algebra. This conjecture seeks a Lie-algebraic description of the algebra of cyclotomic multiple polylogarithms and relates its generators to the K-theory of cyclotomic integer rings. The source does not provide a resolution status for these assertions.