The derivative conjecture for formal multiple Eisenstein series

Let $\mathcal{E}^f$ be the formal multiple Eisenstein-series algebra, let $\mathcal{G}^f$ be the ambient formal algebra, and let $\theta=-\mathcal{D}\circ\varphi$ on the relevant word space. The derivative conjecture for formal multiple Eisenstein series. One should have $\theta=\varphi$ in $\mathcal{E}^f$, so that $\mathcal{E}^f$ is an $\mathfrak{sl}_2$-subalgebra of $\mathcal{G}^f$; moreover, for every $w\in\mathfrak{H}^{\geq2}$,

$$2\pi i\frac{d}{d\tau}G(w)=G(\theta(w)).$$

In particular, $2\pi i\frac{d}{d\tau}$ should be a well-defined derivation on $\mathcal{E}$. The formulas are known in certain formal or classical settings, but the stated identification and full derivation claim remain open.