Dihedral-to-cyclotomic Lie coalgebra conjecture at level one

Let ${\cal D}_{\bullet,\bullet}(N)$ be the bigraded dihedral Lie coalgebra associated with $N$-th roots of unity, and let ${\cal C}_{\bullet,\bullet}(N)$ be the depth-graded cyclotomic Lie coalgebra. The subspace ${\cal D}_{1,1}(1)$ is generated by $\{1\}_1$. Dihedral-to-cyclotomic conjecture. At level one, the natural quotient by this subspace should identify the two bigraded Lie coalgebras:

$${\cal D}_{\bullet,\bullet}(1)/{\cal D}_{1,1}(1)={\cal C}_{\bullet,\bullet}(1).$$

This is a compatibility assertion between the dihedral and cyclotomic descriptions of motivic multiple polylogarithms. The source gives no evidence resolving it.