The prime-level diagonal modular-complex conjecture

Let $p$ be a prime, let $E_m(N)$ be the arithmetic coefficient module defined in the source, let $M^*_{(m)}$ be the dual modular complex, and let ${\rm Gr}\widehat{\cal G}^{(l)}_{\bullet}(\boldsymbol{}[p])$ be the diagonal graded Galois Lie algebra. The prime-level diagonal modular-complex conjecture. There exists a canonical isomorphism between

$$\Bigl(M^*_{(m)}\otimes_{GL_m(\mathbb{Z})}E_m(N)\Bigr)\otimes\mathbb{Q}_l$$

and the depth-$m$ part of the standard cochain complex of ${\rm Gr}\widehat{\cal G}^{(l)}_{\bullet}(\boldsymbol{}[p])$. This proposes an arithmetic modular-complex model for the diagonal Galois Lie algebra at prime level; no resolution is supplied in the text.