The modular-complex description of the graded Galois Lie algebra

Let $L_m$ be the rank-$m$ lattice used to define the modular complex, let $V_m:=L_m\otimes\mathbb{Q}$, let $M^*_{(m)}$ be the dual modular complex, and let $S^{w-m}V_m$ denote the $(w-m)$-th symmetric power. The modular-complex description conjecture. There exists a canonical isomorphism between

$$\Bigl(M^*_{(m)}\otimes_{GL_m(\mathbb{Z})}S^{w-m}V_m\Bigr)\otimes_{\mathbb{Q}}\mathbb{Q}_l$$

and the depth-$m$, weight-$w$ part of the standard cochain complex of ${\rm Gr}\widehat{\cal G}^{(l)}_{\bullet\bullet}$. This would make modular complexes a concrete model for the cohomological pieces of the graded Galois Lie algebra; the supplied text does not state whether the proposed canonical isomorphism is known.