The dihedral Lie algebra equality conjecture for the trivial group

Let ${D}_{\bullet \bullet}:={D}_{\bullet \bullet}(\{e\})$ be the dihedral Lie algebra for the trivial group, let $\xi=\xi_{\{e\}}$ be its embedding, and let ${\rm Gr}{\cal G}^{(l)}_{\bullet \bullet}$ be the corresponding associated graded Galois Lie algebra. The dihedral Lie algebra equality conjecture.

$$\xi({D}_{\bullet \bullet})\otimes \mathbb{Q}_l={\rm Gr}{\cal G}^{(l)}_{\bullet \bullet}.$$

Because the dihedral Lie algebra is defined explicitly, this would give a precise description of the associated graded image of the Galois group. The paper states that the depth $2$ and $3$ cases are proved, while the complete equality remains open.