Descent conjecture for the Hodge-correlator map to the modular complex

Let $E$ be one of the CM elliptic curves $E=\mathbb{C}/\mathbb{Z}[i]$ or $E=\mathbb{C}/\mathbb{Z}[\rho]$, let $\mathfrak{p}$ be the relevant prime ideal, let $T_k$ be the coefficient system, let $\Gamma_1(\mathfrak{p})$ be the corresponding congruence subgroup, and let $M_k^\bullet$ be the modular complex. Let $\theta$ be the map from the modular-complex data to the Chevalley–Eilenberg complex of the depth-$k$ graded piece of the symmetric motivic Lie algebra. Descent conjecture. The map $\theta$ descends to a morphism of complexes

$$\theta:T_k\otimes_{\Gamma_1(\mathfrak{p})}M_k^\bullet\to {\rm CE}^\bullet\left(\operatorname{gr}^D\operatorname{Lie}_{\rm sym}^\vee(E,E[\mathfrak{p}])\right)_{D=k}.$$

Such a descent would extend the established depth-two surjective morphism to higher depth. It requires the second shuffle relations for averaged-base-point Hodge correlators modulo the depth filtration, so it is conditional on the preceding conjectural relation.