The refined dihedral Lie coalgebra description conjecture

Let ${\cal C}(G)$ be the refined dihedral Lie coalgebra for a finite commutative group $G$, let ${C}_{\bullet}(G)$ be its graded dual, and let ${\rm Gr}^W{\cal G}^{(l)}$ be the weight-graded Galois Lie algebra. The refined dihedral description conjecture. There is a canonical isomorphism

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

Consequently, ${\cal G}^{(l)}$ and ${C}_{\bullet}\otimes\mathbb{Q}_l$ are noncanonically isomorphic. This supplies a refined, rather than merely associated-graded, dihedral model for the Galois Lie algebra; the supplied text gives no resolution.