The conjecture for multiple Eisenstein-diamond series

Let $\mathfrak{H}^0$ be the admissible word space, let $\mathfrak{H}^0_\ast$ denote its harmonic-product algebra, and define $G^\diamondsuit=G\circ\mathcal{D}:\mathfrak{H}^0\to\mathcal{E}$. The conjecture for multiple Eisenstein-diamond series. The map $G^\diamondsuit$ should be an algebra homomorphism; for every $w\in\mathfrak{H}^0$,

$$2\pi i\frac{d}{d\tau}G^\diamondsuit(w)=G^\diamondsuit(w\mathbin{{{{{{{{{\shuffle}}}}}}}}}z_2-w\ast z_2);$$

and all its relations should be generated by Drop1 relations and products:

$$\ker G^\diamondsuit=\mathsf{Drop1}_\ast,$$

where

$$\mathsf{Drop1}_\ast=\operatorname{Span}_\mathbb{Q}\{(\mathcal{D}(u)-u)\ast v\mid u,v\in\mathfrak{H}^0\}.$$

This proposes an intrinsic algebraic and differential description of the multiple Eisenstein-diamond series and remains open.