The modular-complex conjecture for cyclotomic Galois Lie algebras

Let $M^*_{(m)}$ be the modular complex, let $\Gamma_1(m;N)$ be the indicated congruence subgroup, let ${\rm V}_m$ be the representation appearing in the source, and let $\Lambda^*_{(m,w)}$ denote the depth-$m$, weight-$w$ part of the standard cochain complex. Let $\mu^*_{m;w}$ be the map from the modular complex to the corresponding cochain complex of the graded Galois Lie algebra. Modular-complex conjecture. If $N=1$, or if $N=p$ is prime and $w=m$, then the map

$$M^*_{(m)}\otimes_{\Gamma_1(m;N)}S^{w-m}{\rm V}_m\longrightarrow\Lambda^*_{(m,w)}\operatorname{Gr}{\mathcal G}^{(l)}_{\bullet\bullet}(\mu_N)^\vee$$

is an isomorphism. This predicts that modular complexes compute the indicated cochain complexes in the stated level and weight cases; no resolution evidence is supplied.