The diagonal dihedral Lie algebra conjecture for prime cyclotomic level

Let $p$ be a prime number, let $\boldsymbol{}[p]$ be the group of $p$-th roots of unity, let ${D}^{\Delta}_{\bullet}(\boldsymbol{}[p])$ be the diagonal, depth-equals-weight part of the dihedral Lie algebra, and let ${\rm Gr}{\cal G}^{(l)}_{\bullet}(\boldsymbol{}[p])$ be the diagonal Galois Lie algebra. The diagonal dihedral Lie algebra conjecture.

$${\rm Gr}{\cal G}^{(l)}_{\bullet}(\boldsymbol{}[p])=\xi_{\boldsymbol{}[p]}({D}^{\Delta}_{\bullet}(\boldsymbol{}[p])).$$

This is the prime-level diagonal analogue of the full dihedral description. The paper says that the conjecture is proved in depths $2$ and $3$, with higher-depth cases remaining open.