The derivative and relations conjecture for multiple Eisenstein series

Let $\mathfrak{H}^{\geq2}=\mathbb{Q}\langle z_k\mid k\geq2\rangle$ be the noncommutative word space, let $G:\mathfrak{H}^{\geq2}\to\mathcal{E}$ send $z_{k_1}\cdots z_{k_r}$ to $G_{k_1,\ldots,k_r}(\tau)$, and let $\theta=-\mathcal{D}\circ\varphi$ be the stated $\mathbb{Q}$-linear operator. For $R(u,v)$ as defined in the paper, set

$$\mathsf{DR}_\ast=\operatorname{Span}_\mathbb{Q}\{\theta^n(R(u,v))\ast w\mid n\geq0,u,v,w\in\mathfrak{H}^{\geq2}\}.$$

The derivative and relations conjecture for multiple Eisenstein series. For every $w\in\mathfrak{H}^{\geq2}$,

$$2\pi i\frac{d}{d\tau}G(w)=G(\theta(w))=G(\mathcal{D}(w\mathbin{{{{{{{{{\shuffle}}}}}}}}}z_2))-G_2G(w),$$

and

$$\mathsf{DR}_\ast=\ker G.$$

Thus the proposed derivative formula and the generated family of relations should describe all linear relations among multiple Eisenstein series. These assertions are presented as the main conjecture of the paper and remain open.