The dihedral Lie algebra containment conjecture for cyclotomic Galois symmetries

Let $N$ be a positive integer, let $\boldsymbol{}[N]$ denote the group of $N$-th roots of unity, let ${\rm Gr}{\cal G}_{\bullet \bullet}^{(l)}(\boldsymbol{}[N])$ be the associated graded Lie algebra of the level-$N$ $l$-adic Galois action, let ${D}_{\bullet \bullet}(\boldsymbol{}[N])$ be the dihedral Lie algebra, and let $\xi_{\boldsymbol{}[N]}$ be the displayed embedding into the graded derivation Lie algebra. The dihedral Lie algebra containment conjecture.

$${\rm Gr}{\cal G}_{\bullet \bullet}^{(l)}(\boldsymbol{}[N]) \subset \xi_{\boldsymbol{}[N]}({D}_{\bullet \bullet}(\boldsymbol{}[N])) \otimes_{\mathbb{Q}} \mathbb{Q}_l.$$

This asserts that the power-shuffle and the other structural relations described in the paper constrain the Galois Lie algebra to the explicitly constructed dihedral subalgebra; only special depth quotients are proved in the paper, so the full containment remains open.